ANALYTIC GEOMETRY
I
1. ECE Board November 1995 The semi-major axis of an ellipse is 4 and its semi-minor axis is 3. The distance from the center to the directrix is A. 6.047 B. 6.532 . 0.6614 !. 6.222
A. ,11-24 ,11-24 B. ,-11-20 . ,11-1( !. ,11-20 7. ECE Board April 1998 !etermine B s$ch that 3x 2& 7 ' 0 is perpendic$lar to 2x B& 2 ' 0. A. 5 B. 4 . 3 !. 2
2. ECE Board April 1995/ March 1996, April 1999 "ind the e#$ation of the directrix of the para%ola & 2 ' 16x. A. x ' -4 B. x ' -( . x ' 4 !. x ' (
(. ECE Board April 1998 "ind the al$e of for *hich the e#$ation x2 &2 4x 2& ' 0 represents represents a point circle. A. 5 B. 6 . -6 !. -5
3. ECE Board November 1997 The midpoint of the line se)ment %et*een +1,x& and +2,-24 is +m,2-1. "ind the coordinates of + 1. A. ,6-6 B. ,6-5 . ,5-6 !. ,-66
. ECE Board April 1998 The diameter of a circle descri%ed %& x2 &2 ' 16 is A. 43 B. 16 . (3 !. 4
4. ECE Board November 1997 /ien the ellipse ,x 236 ,&232 ' 1. !etermine the distance %et*een the foci. A. ( B. 4 . 2 !. 3
10. ECE Board April 1998 "ind the e#$ation of the axis of s&mmetr& of the f$nction & ' 2x 2 7x 5. A. 7x 4 ' 0 B. 4x 7 ' 0 . 4x 7 ' 0 !. 4x 2 ' 0
5. ECE Board November 1997 "ind the coordinates of the point +,24 *ith respect to the translated axis *ith ori)in at ,13. A. ,1-1 B. ,-1-1 . ,11 !. ,-11
11. ECE Board April 1998 +oint +,x& moes *ith a distance from point ,01 one-half of its distance from line & ' 4. The e#$ation of its loc$s is A. 2x2 4&2 ' 5 B. 4x2 3&2 ' 12 . 2x2 5&3 ' 3 !. x2 2&2 ' 4
6. ECE Board Exam April 1998 The se)ment from ,-14 to ,2-2 is extended three times its o*n len)th. The terminal point is
ENGINEERING MATHEMATICS
E2 - 1
I
ANALYTIC GEOMETRY
I
12. ECE Board April 1998 The major axis of the elliptical path in *hich the earth moes aro$nd the s$n is approximatel& 1(6000000 1(6000000 miles and the eccentricit& of the ellipse is 160. !etermine the apo)ee of the earth. A. 3000000 3000000 miles B. 1450000 miles . 4335100 miles !. 4550000 miles
A points moe so that that its distance from the point ,2-1 is e#$al to its distance from the x-axis. The e#$ation of the loc$s is A. x2 4x 2& 5 ' 0 B. x2 4x 2& 5 ' 0 . x2 4x 2& 5 ' 0 !. x2 4x 2& 5 ' 0 1(. ECE Board November 1999 The point of intersection of the planes x 5& 28 ' 3x 2& 8 ' 3 and x & 8 ' 2 is at A. ,121 B. ,21-1 . ,1-12 !. ,-1-12
13. ECE Board November 1998 A line passes thro$)h thro$)h point ,22. ,22. "ind the e#$ation of the line if the len)th of the line se)ment intercepted %& the coordinate axes is the s#$are root of 5. A. 2x & 2 ' 0 B. 2x & 2 ' 0 . 2x & 2 ' 0 !. 2x & 2 ' 0
1. ECE Board November 1999/ April 2005/ ECE Board Board April April 200 /ien the points ,37 and ,-4-7. 9ole the distance %et*een them. A. 15.65 B. 17.65 . 16.65 !. 14.65
14. ECE Board November 1998 "ind the area of the trian)le *hich the line 2x 3& 6 ' 0 form *ith the coordinate axes. A. 3 B. 4 . 5 !. 2
20. ECE Board November 1999 "ind the distance of directrix from f rom the center of an ellipse if its major axis is 10 and its minor axis is (. A. (.5 B. (.1 . (.3 !. (.7
15. ECE Board November 1998 !etermine the coordinates of the point *hich is three-fifths of the *a& from the point ,2-5 to the point ,-35. A. ,-11 B. ,-2-1 . ,-1.-2 !. ,1-1
21. ECE Board April 2000 "ind the coordinates of the ertex of the para%ola & ' x 2 4x 1 %& main) $se of the fact that at the ertex the slope of the tan)ent is 8ero. A. ,2-3 B. ,-2-3 . ,-1-3 !. ,3-2
16. ECE Board April 1999 f the points ,-23 ,x& and ,-35 lie on a strai)ht line then the e#$ation of the line is . A. x 2& 1 ' 0 B. 2x & 1 ' 0 . x 2& 1 ' 0 !. 2x & 1 ' 0
22. ECE Board April 2000 17. ECE Board November 1999 ENGINEERING MATHEMATICS
E2 - 2
I
ANALYTIC GEOMETRY
I
"ind the area of the hexa)on AB!:" formed %& joinin) the points A,14 B,0-3 ,23 !,-12 :,-2-1 and ",30. A. 24 B. 20 . 22 !. 15 23. ECE Board April 2000 The para%olic antenna has an e#$ation &2 (x ' 0. !etermine the len)th of the lat$s rect$m. A. ( B. 10 . 12 !.
. 13 !. 6 2(. ECE Board November 2001 "ind the an)le %et*een the planes 3x & 8 5 ' 0. A. 62.45= B. 52.45= . (2.45= !. 72.45= 2. ECE Board November 2001 "ind the e#$ation of a line *here xintercept is 2 and &-intercept is -2. A. 2x 2& 2 ' 0 B. x & 2 ' 0 . 2& 2x 2 ' 0 !. x & 1 ' 0
24. ECE Board November 2000 A line 4x 2& 2 ' 0 is coincident *ith the line A. 4x 4& 2 ' 0 B. 4x 3& 3 ' 0 . (x 4& 2 ' 0 !. (x 4& 4 ' 0
30. ECE Board April 2002 "ind the al$e of if the distance from the point ,21 to the line 5x 12& ' 0. is 2. A. 5 B. 2 . 4 !. 3
25. ECE Board April 2001 "ind the e#$ation of the para%ola *hose axis is parallel to the x-axis and passes thro$)h the points ,31 ,00 and ,(-4. A. x2 2x & ' 0 B. x2 2x & ' 0 . &2 2& x ' 0 !. &2 2& x ' 0
31. ECE Board April 2002 !etermine the farthest distance from the point ,37 to the circle x 2 &2 4x 6& 12 ' 0. A. 6.40 B. 1.40 . 11.40 !. 4.60
26. ECE Board April 2001/ November 2002 The directrix of a para%ola is the line & ' 5 and its foc$s is at the point ,4-3. ;hat is the len)th of lat$s rect$m< A. 1( B. 14 . 16 !. 12
32. ECE Board November 2002 "ind the e#$ation of the perpendic$lar %isector of the line joinin) ,40 and ,63. A. 4x 6& 2 ' 0 B. 4x 6& 2 ' 0 . 4x -6& 2 ' 0 !. 4x 6& 2 ' 0
27. ECE Board November 2001 A point +,x2 is e#$idistant from the points ,-2 and ,4-7. The al$e of x is 33. ECE Board April 200 A line has an e#$ation of x 5& 5 ' 0. "ind the e#$ation of the line
A. 113 B. 203 ENGINEERING MATHEMATICS
E2 - 3
I
ANALYTIC GEOMETRY
I
thro$)h point ,31 that is parallel to this line. A. x 6& ' 0 B. x 5& ( ' 0 . x 7& ( ' 0 !. x 3& 5 ' 0
3. ECE Board November 200 ;hat is the e#$ation of the circle *ith center at the ori)in and a radi$s of 5< A. x2 &2 ' 1 B. x2 &2 ' 25 . x2 &2 ' 10 !. x2 &2 ' 5
34. ECE Board April 200 !etermine the ertex of the para%ola & ' -x2 (x 2. A. ,1(4 B. ,-4-1( . ,41( !. ,-41(
40. ECE Board November 200 ;hat is the e#$ation of the line thro$)h ,-35 *hich maes an an)le of 45 de)rees *ith the line 2x & ' 12< A. x 3& 12 ' 0 B. x 3& 1( ' 0 . x 2& 7 ' 0 !. x 3& 1( ' 0
35. ECE Board April 200 ;hat is the e#$ation of a circle *ith its center at the ori)in and if the point ,11 lies on the circ$mference of the circle< A. ,x12 ,&12 ' 2 B. ,x12 ,&12 ' 4 . x2 &2 ' 2 !. x2 &2 ' 4
41. ECE Board November 200 !etermine the ac$te an)le %et*een the lines & 3x ' 2 and & 4x ' . A. 4.3 de) B. 3.75 de) . 5.35 de) !. 2.53 de)
36. ECE Board April 200 ;hat is the distance of the line 4x 3& 5 ' 0 from the point ,42< A. 5 B. 4 . 2 !. 3
42. ECE Board November 200 !etermine the e#$ation of the perpendic$lar %isector of the se)ment +> if +,-23 and >,4-5. A. 3& 3x 7 ' 0 B. 4x 3& 7 ' 0 . 6x (& 14 ' 0 !. 3x 4& 7 ' 0
37. ECE Board April 200 f the lines 4x & 2 ' 0 and x 2& 1 ' 0 are perpendic$lar to each other determine the al$e of A. 3 B. 4 . 1 !. 2 3(. ECE Board April 200 A trian)le is dra*n *ith ertices at ,-1-1 ,13 and ,41. ;hat is the median from ertex ,41<
43. ECE Board April 200! "ind the ol$me of the p&ramid formed in the first octant %& the plane 6x 10& 58 30 ' 0 and the coordinate axes. A. 13 B. 12 . 14 !. 15
A. 10 $nits B. 4 $nits . 5 $nits !. 6 $nits
44. ECE Board April 200! A circle *ith its center in the first #$adrant is tan)ent to %oth x and & axes. f its radi$s is 4 *hat is the e#$ation of the circle<
ENGINEERING MATHEMATICS
E2 - 4
I
ANALYTIC GEOMETRY
I
A. ,x42 ,&42 ' 16 B. ,x(2 ,& (2 ' 16 . ,x 42 ,&42 ' 16 !. ,x42 ,&4 2 ' 16
The distance from a point ,2& to a line 4x 3& 7 ' 0 is e#$al to 5. "ind the al$e of &. A. 12 B. ( . 5 !. 7
45. ECE Board April 200! A circle is descri%ed %& the e#$ation x2 &2 16x ' 0. ;hat is the len)th of the chord *hich is 4 $nits from the center of the circle< A. 6.3 $nits B. 13.(6 $nits . 11.55 $nits !. .(5 $nits
51. #roblem$ The distance from a point ,2& to a line x 2& 3 ' 0 e#$al to 5 . "ind the al$e of &. A. & ' 5 B. & ' -7 . & ' !. 7 ' 3
46. ECE Board April 200! ;hat is the e#$ation of the line that passes thro$)h ,-35 and is parallel to the line 4x -2& 2 '0< A. 4x 2& 22 ' 0 B. 2x & 10 ' 0 . 4x 2& 11 ' 0 !. 2x & 11 ' 0
52. #roblem$ "ind the distance %et*een the points ,25 and the line x 2& 3 ' 0. A. 5
47. ECE Board April 200! ;hat is the distance %et*een line x 2& ( ' 0 and the point ,5-2< A. 4.20 B. 4.44 . 4.02 !. 4.22
B.
(
.
6
!.
53. #roblem$ "ind x if the distance %et*een points ,x4 and ,34 is e#$al to 10. A. 13-7 B. 12-6 . 11-4 !. 14-7
4(. ECE Board April 2005/ April 1999 /ien t*o points ,-4-7 and ,37. ;hat is the distance %et*een them< A. 15.65 B. 4.5( . 245 !. 1
54. #roblem$ "ind the distance %et*een the lines 3x 4& 12 ' 0 and 3x 4& 22 ' 0. A. 3 B. 4 . 1 !. 2
4. CE Board Ma" 1992 "ind the distance %et*een the )ien lines 4x 3& ' 12 and 4x 3& ' -(.
55. #roblem$ "ind the distance coordinate of the center of the circle 2x2 (x 2&2 12& ' 1 and the x and & axes. A. ,-23 B. ,23 . ,2-3 !. ,-2-3
A. 4 B. 10 . ( !. 3 50. #roblem$ ENGINEERING MATHEMATICS
E2 - 5
I
ANALYTIC GEOMETRY
I hain) a slope of ? *hich passes thr$ the intersection of the lines. A. 2x 4& 11 ' 0 B. 2x (& 12 ' 0 . 2x 6& 13 ' 0 !. 2x 5& ' 0
56. #roblem$ "ind the slope of the line *hose parametric e#$ations are x ' 2 t and & ' 1 2t. A. -2 B. 2 . ? !. @
62. #roblem$ T*o lines hae an e#$ation of 2x & 2 ' 0 and 2x & 4 ' 0. ; hat is the e#$ation of the line %isectin) the %i))er an)le formed %& the intersection of the lines. A. & 5 ' 0 B. & 3 ' 0 . & 3 ' 0 !. & 5 ' 0
57. #roblem$ A line has a parametric e#$ation of x ' 4 3t and & ' 7 t. "ind the &intercept of the line. A. 143 B. 173 . 163 !. 317
63. #roblem$ The points ,13 and ,55 are t*o opposite ertices of a rectan)le. The other t*o ertices lie on the line & ' 2x . "ind the coordinates of the centroid of the rectan)le. A. & ' 4 B. & ' ( . & ' 2 !. & ' 0
5(. #roblem$ A line has a parametric e#$ation of x ' 4 3t and & ' 7 t. "ind the distance from the ori)in to this line. A. 6.2( B. 5.3( . 10.76 !. 2.17 5. #roblem$ A line has a parametric e#$ation of x ' 4 3t and & ' 7 t. "ind the an)le in de)rees %et*een this line and the x-axis. A. 16.43= B. 17.43= . 1(.43= !. 1.43=
64. #roblem$ The points ,13 and ,55 are t*o opposite ertices of a rectan)le. The other t*o ertices lie on the line & ' 2x . "ind the al$e of . A. ' -2 B. ' 2 . ' 1 !. ' -1
60. #roblem$ T*o lines hae an e#$ation of 2x & 2 ' 0 and 2x & 4 ' 0. "ind the smallest an)le %et*een the t*o lines. 65. #roblem$ The points ,13 and ,55 are t*o opposite ertices of a rectan)le. The other t*o ertices lie on the line & ' 2x . "ind the area of the rectan)le. A. 5 s#. $nits B. 6 s#. $nits . 7 s#. $nits !. ( s#. $nits
A. 63.13= B. 0= . 53.13= !. 45= 61. #roblem$ T*o lines hae an e#$ation of 2x & 2 ' 0 and 2x & 4 ' 0. !etermine the e#$ation of the line ENGINEERING MATHEMATICS
E2 - 6
I
ANALYTIC GEOMETRY
I B. 5x & 10 ' 0 . 5x & 14 ' 0 !. x 5& 14 ' 0
66. #roblem$ T*o lines hain) an e#$ation of 4x 3& 11 ' 0 and 5x 12& 2 ' 0 intersect each other. "ind the e#$ation of the line %isector of the smaller an)le formed %& the intersection of the t*o lines. A. 3x 17& ' 11 B. 3x 11& ' 17 . 11x 3& ' 17 !. 11x 17& ' 3
71. #roblem$ A circle has its center at ,3-2 is tan)ent to the line 3x 4& 26 ' 0. omp$te the e#$ation of the circle. A. x2 &2 6x 4& ' 12 B. x2 &2 6x 4& ' 12 . x2 &2 6x 4& ' 12 !. x2 &2 6x 4& ' 12
67. #roblem$ T*o lines hain) an e#$ation of 4x 3& 11 ' 0 and 5x 12& 2 ' 0 intersect each other. "ind the smaller an)le %et*een the t*o lines. A. 75=45 B. 45=75 . 60=75 !. 75=60
72. #roblem$ A circle has its center at ,3-2 is tan)ent to the line 3x 4& 26 ' 0. omp$te the e#$ation of the normal. A. 3x 2& ' 12 B. 4x 3& ' 1( . 4x 4& ' 12 !. 3x 3& ' 11 73. #roblem$ A circle has its center at ,3-2 is tan)ent to the line 3x 4& 26 ' 0. omp$te the point of tan)enc& of the circle. A. 42 B. 34 . 62 !. 26
6(. #roblem$ T*o lines hain) an e#$ation of 4x 3& 11 ' 0 and 5x 12& 2 ' 0 intersect each other. "ind the e#$ation of the line perpendic$lar to the line %isector of an)le formed %& intersection of the t*o lines *hich passes thr$ the intersection of the t*o lines. A. 11x 5& ' 1 B. 11x 6& ' . 11x 3& ' 1 !. 11x 6& ' 6 6. #roblem$ ;hat is the e#$ation of the line hain) a slope of 2 and passin) thro$)h the point ,-1 1.
74. #roblem$ T*o circles hae e#$ations of x2 &2 4x 4& 4 ' 0 and x2 &2 4x (& 4 ' 0. "ind the distance %et*een the centers of the t*o circles. A. 6 B. 4 . 3 !. 2
A. 2x & 3 ' 0 B. 3x & 3 ' 0 . 2x & 3 ' 0 !. 3x & 3 ' 0 70. #roblem$ A line has an e#$ation of x 5& 5 ' 0. "ind the e#$ation of the line thro$)h points ,31 that is perpendic$lar to this line. A. x 5& 4 ' 0
75. #roblem$
ENGINEERING MATHEMATICS
E2 - 7
I
ANALYTIC GEOMETRY
I
T*o circles hae e#$ations of x2 &2 4x 4& 4 ' 0 and x2 &2 4x (& 4 ' 0. !etermine the e#$ation of the radical axis. A. & ' 2 B. & ' 0 . & ' 4 !. & ' 3
& 10 ' 0 and 2x & 2 ' 0. "ind the area of the trian)le circ$mscri%in) the circle. A. 25 s#.$nits B. 15 s#.$nits . 30 s#.$nits !. 45 s#.$nits (1. #roblem$ A trian)le has its sides hain) e#$ation e#$al to x 2& 5 ' 0 2x & 10 ' 0 and 2x & 2 ' 0. "ind the e#$ation of the circle inscri%ed in the trian)le. A. x2 &2 4x 2& ' 0 B. 3x2 3&2 3x 2& ' 0 . 4x2 4&2 x 4& ' 0 !. 2x2 2&2 -2x 3& ' 0
76. #roblem$ T*o circles hae e#$ations of x2 &2 4x 4& 4 ' 0 and x2 &2 4x (& 4 ' 0. omp$te the len)th of the common external tan)ent. A. 5.66 B. 6.74 . 4.12 !. 6.43
(2. #roblem$ A trian)le has its sides hain) e#$ation e#$al to x 2& 5 ' 0 2x & 10 ' 0 and 2x & 2 ' 0. "ind the area of the circle inscri%ed in the trian)le. A. 17.51 s#.$nits B. 15.67 s#.$nits . 13.54 s#.$nits !. 15.71 s#.$nits
77. #roblem$ A circle is circ$mscri%in) a trian)le formed %& the lines & ' 0 & ' x and 2x 3& ' 10. "ind the area of the trian)le inscri%ed in the circle. A. 7 B. 3 . ( !. 5 7(. #roblem$ A circle is circ$mscri%in) a trian)le formed %& the lines & ' 0 & ' x and 2x 3& ' 10. "ind the e#$ation of the circle. A. x2 &2 5x & ' 0 B. 2x2 &2 3x 2& ' 0 . x2 3&2 x & ' 0 !. 2x2 &2 5x & ' 0
(3. #roblem$ 7. #roblem$ A circle is circ$mscri%in) a trian)le formed %& the lines & ' 0 & ' x and 2x 3& ' 10. "ind the area of the circle. A. 1(.23 s#.$nits B. 20.42 s#.$nits . 22.23 s#.$nits !. 35.33 s#. $nits
A circle has an e#$ation of x2 &2 ' 4(. !etermine the area of the se)ment of the circle c$t %& the line joinin) the intersection of the circle and the c$re x2 (& ' 0. A. 12.5 B. 11.32 . 13.67 !. 17. 54
(0. #roblem$ A trian)le has its sides hain) e#$ation e#$al to x 2& 5 ' 0 2x
(4. #roblem$
ENGINEERING MATHEMATICS
E2 - 8
I
ANALYTIC GEOMETRY
I
A circle has an e#$ation of x2 &2 ' 4(. !etermine the area of the se)ment of the circle c$t %& the line joinin) the intersection of the circle and the c$re x2 (& ' 0. A. 24.45 s#.$nits B. 23.75 s#.$nits . 23.24 s#.$nits !. 27.12 s#.$nits
A. x 4 ' 0 B. x 2 ' 0 . x 4 ' 0 !. x 4 ' 0 0. #roblem$ A para%ola has an e#$ation of x2 ' 16&. omp$te the a%scissa of a point B on the c$re *hich has its ordinate e#$al to 4. A. 4 B. 2 . ( !. 16
(5. #roblem$ A circle has an e#$ation of x2 &2 ' 4(. omp$te the common area %et*een the circle x2 &2 ' 4( and the c$re x2 (& ' 0. A. 57.23 s#.$nits B. 77.54 s#.$nits . 53.43 s#.$nits !. 67.(5 s#.$nits
1. #roblem$ A para%ola hain) its axis alon) the x-axis passes thro$)h ,-36 if the ertex is at the ori)in. omp$te the e#$ation of the para%ola. A. &2 ' -12x B. &2 ' -14x . &2 ' -16x !. &2 ' -(x
(6. #roblem$ A para%ola has an e#$ation of x2 ' 20&. omp$te the lat$s rect$m of the para%ola. A. 25 lat$s rect$m B. 23.5 lat$s rect$m . 20 lat$s rect$m !. 1( lat$s rect$m
2. #roblem$ A para%ola hain) its axis alon) the x-axis passes thro$)h ,-36 if the ertex is at the ori)in. ocate the coordinates of the foc$s. A. ",-30 B. ",03 . ",0-3 !. ",-33
(7. #roblem$ A para%ola has an e#$ation of x2 ' 20&. ocate the coordinates of the foc$s of the para%ola. A. ,50 B. ,05 . ,45 !. ,54
3. #roblem$ A para%ola has an e#$ation of x2 4x 16& ' 44. ocate the coordinates of the ertex of the para%ola. A. ,-23 B. ,3-2 . ,4-3 !. ,3-3
((. #roblem$ A para%ola has an e#$ation of x2 ' 20&. !etermine the e#$ation of the directrix of the para%ola. A. & 5 ' 0 B. & 5 ' 0 . & 20 ' 0 !. & 20 ' 0
4. #roblem$ A para%ola has an e#$ation of
(. #roblem$ A para%ola has an e#$ation of x2 ' 16&. !etermine the e#$ation of the directrix. ENGINEERING MATHEMATICS
E2 - 9
I
ANALYTIC GEOMETRY
I
x2 4x 16& ' 44. ocate the coordinates of the foc$s of the para%ola. A. ",-2-1 B. ",-2-2 . ",-31 !. ",22
. 2 !. 1 100. #roblem$ A point moes so that its distance from point ,2-1 is e#$al to its distance from the x-axis. omp$te the e#$ation of the directrix. A. & 1 ' 0 B. & 1 ' 0 . & 2 ' 0 !. & 2 ' 0
5. #roblem$ A para%ola has an e#$ation of x2 4x 16& ' 44. !etermine the e#$ation of the directrix of the para%ola. A. & 5 ' 0 B. & 4 ' 0 . & 6 ' 0 !. & 4 ' 0
101. #roblem$ An ellipse has an e#$ation e#$al to x2 144x 16&2 6& 45 ' 0. omp$te the center of the c$re. A. ,(4 B. ,47 . ,(-3 !. ,3-(
6. #roblem$ A para%ola has an e#$ation of &2 4& 4x ( ' 0. ocate the ertex of the para%ola. A. ,-44 B. ,53 . ,4-1 !. ,-3-2
102. #roblem$ An ellipse has an e#$ation e#$al to x2 144x 16&2 6& 45 ' 0. omp$te the eccentricit& of the c$re. A. 0.556 B. 0.661 . 0.(41 !. 0.6(
7. #roblem$ A para%ola has an e#$ation of &2 4& 4x ( ' 0. ocate the foc$s of the para%ola. A. ",-3-2 B. ",-2.2 . ",-2-2 !. ",4-4 (. #roblem$ A point moes so that its distance from point ,2-1 is e#$al to its distance from the x-axis. "ind the e#$ation of the loc$s. A. x2 5x 2& 5 ' 0 B. x2 4x 2& 5 ' 0 . x2 3x 2& 5 ' 0 !. x2 4x 2& 5 ' 0
103. #roblem$ An ellipse has an e#$ation e#$al to x2 144x 16&2 6& 45 ' 0. omp$te the len)th of the lat$s rect$m. A. 7.5 B. 3.6 . 4.5 !. .3
. #roblem$ A point moes so that its distance from point ,2-1 is e#$al to its distance from the x-axis. omp$te the len)th of lat$s rect$m. A. 5 B. 3
104. #roblem$ An ellipse has an e#$ation e#$al to x2 4&2 72x 24& 144 ' 0. omp$te the location of its ertices. A. ,64 and ,56
ENGINEERING MATHEMATICS
E2 - 10
I
ANALYTIC GEOMETRY
I
B. ,40 and ,46 . ,5 and ,6 !. ,(3 and ,74
!. 0.732 110. #roblem$ The distance %et*een the foci of an ellipse is e#$al to ( and the second eccentricit& is e#$al to 1.333. omp$te the len)th of lat$s rect$m. A. 4.( B. 3.1 . 2.5 !. 3.6
105. #roblem$ An ellipse has an e#$ation e#$al to x2 4&2 72x 24& 144 ' 0. omp$te the distance %et*een the foci. A. 6.423 B. 6.733 . 4.472 !. (.135
111. #roblem$ The distance %et*een the foci of an ellipse is e#$al to ( and the second eccentricit& is e#$al to 1.333. omp$te the distance %et*een the directrices. A. 12.5 B. 17.4 . 13.5 !. 16.42
106. #roblem$ An ellipse has an e#$ation e#$al to x2 4&2 72x 24& 144 ' 0. omp$te the distance from the center to one of its directrix. A. 4.025 B. 7.421 . 6.774 !. 3.57(
112. #roblem$ The distance %et*een the foci of an ellipse is e#$al to ( and the second eccentricit& is e#$al to 1.333. omp$te the perimeter of the c$re. A. 26.( B. 22.4 . 25.1 !. 1.43
107. #roblem$ An ellipse has an eccentricit& of 13 and the distance %et*een the foci is e#$al to 4. omp$te the len)th of lat$s rect$m. A. 77.4 B. 14.( . 6.7( !. 10.67 10(. #roblem$ An ellipse has an eccentricit& of 13 and the distance %et*een the foci is e#$al to 4. omp$te the distance from the farthest ertex to one of its directrix. A. 12 B. 16 . 24 !. 32
113. #roblem$ The len)th of the lat$s rect$m of an ellipse is 45 of the len)th of its minor axis. f the distance %et*een their ertices is 20 comp$te its eccentricit&. A. 0.60 B. 0.73 . 0.31 !. 0.56
10. #roblem$ An ellipse has an eccentricit& of 13 and the distance %et*een the foci is e#$al to 4. omp$te the second eccentricit& of ellipse. A. 0.644 B. 0.353 . 0.321
114. #roblem$ The len)th of the lat$s rect$m of an ellipse is 45 of the len)th of its minor axis. f the distance %et*een their ertices is 20 comp$te the distance %et*een the directrices.
ENGINEERING MATHEMATICS
E2 - 11
I
ANALYTIC GEOMETRY
I
A. 66.67 B. 33.33 . 45.66 !. 23.33
The len)th of the lat$s rect$m of a h&per%ola is e#$al to 1( and the distance %et*een the foci is 12. omp$te the e#$ation of the as&mptote of the h&per%ola. A. & = 3x
115. #roblem$ The distance from point A
(
)
6 cos θ 2 sin θ to the center of
ellipse is e#$al to 2. f the e#$ation of ellipse 2x2 6&2 ' 12. "ind the al$e of C. A. 60= B. 15= . 45= !. 74=
= &= &=
B. &
4x
.
5x
!.
6x
120. #roblem$ A h&per%ola has a len)th of lat$s rect$m e#$al to 1 and slope of as&mptotes is E 12. "ind the e#$ation of the h&per%ola. A. x2 4&2 ' 6 B. x2 4&2 ' 4 . x2 4&2 ' 3 !. x2 4&2 ' 4
116. #roblem$ A h&per%ola has an e#$ation of 16&2 x2 36x 6& 36 ' 0. omp$te the coordinates of the center of the c$re. A. ,2-3 B. ,-2.-3 . ,-32 !. ,36
121. #roblem$ A h&per%ola has a len)th of lat$s rect$m e#$al to 1 and slope of as&mptotes is E 12. omp$te the distance %et*een the directrices of the h&per%ola. A. 4.6( B. 3.5( . (.65 !. 5.57
117. #roblem$ A h&per%ola has an e#$ation of 16x2 &2 12(x 0& 113 ' 0. ocate the position of the ertices of the c$re.
122. #roblem$ A h&per%ola passes thro$)h ,20 and *hose foci are ,-40 and ,40. "ind the e#$ation of the h&per%ola. A. 3x2 &2 ' 12 B. x2 3&2 ' 12 . 3x2 &2 ' 12 !. x2 3x2 ' 12
A. D,56 and D,1-5 B. D,7-5 and D,5-1 . D,7-7 and D,15 !. D,7-5 and D,1-5) 11(. #roblem$ The len)th of the lat$s rect$m of a h&per%ola is e#$al to 1( and the distance %et*een the foci is 12. "ind the e#$ation of the c$re if the conj$)ate axis is parallel to the &axis. A. 3x2 &2 ' 27 B. 4x2 3&2 ' 21 . 3x2 &2 ' 27 !. 3x2 3&2 ' 2
123. #roblem$ A point moes so that the difference %et*een its distances from ,05 and ,0-5 is (. "ind the e#$ation of the loc$s of the point. A. &2 16x2 ' 144 B. 3&2 4x2 ' 12 . 4&2 3x2 ' 144 !. &2 16x2 ' 12
11. #roblem$ ENGINEERING MATHEMATICS
E2 - 12
I
ANALYTIC GEOMETRY
I
124. #roblem$ T*o )ro$nd stations are located %& its coordinates as A,00 and B,05 the $nit %ein) 1 m. An airplane pilot cond$ctin) a reconnaissance s$re& no*s from the radar that at a certain instant he is 3 m. nearer B than A. ;hat is the e#$ation of the c$re that defines this data< A. x2 16&2 (0& 64 ' 0 B. x2 16&2 & 16 ' 0 . x2 16&2 (0& 64 ' 0 !. x2 16&2 (0& 75 ' 0
12(. #roblem$ An e#$ilateral h&per%ola has an e#$ation of x 2 &2 ' . omp$te the location of the ertices. A. D,30 and ,-30 B. D,-30 and ,03 . D,-33 and ,30 !. D,-32 and ,32 12. #roblem$ An e#$ilateral h&per%ola has an e#$ation of x 2 &2 ' . omp$te the eccentricit& of the e#$ilateral h&per%ola. A. 2 B. 1.414 . 3.14 !. 5.34
125. #roblem$ T*o )ro$nd stations are located %& its coordinates as A,00 and B,05 the $nit %ein) 1 m. An airplane pilot cond$ctin) a reconnaissance s$re& no*s from the radar that at a certain instant he is 3 m. nearer B than A. omp$te the eccentricit& of this c$re. A. 1.(6 B. 1.67 . 1.53 !. 1.7
130. #roblem$ A c$re has an e#$ation of x2 ' 16&. omp$te the e#$ation of the tan)ent at point ,41. A. x ' 2& 2 B. x ' 5& 2 . x ' 4& 1 !. x ' & 4
126. #roblem$ T*o )ro$nd stations are located %& its coordinates as A,00 and B,05 the $nit %ein) 1 m. An airplane pilot cond$ctin) a reconnaissance s$re& no*s from the radar that at a certain instant he is 3 m. nearer B than A. omp$te the len)th of the lat$s rect$m. A. 7.53 B. 6.44 . 5.33 !. 11.34
131. #roblem$ A c$re has an e#$ation of x2 ' 16&. omp$te the e#$ation of the normal at point ,41. A. 2x & ' 0 B. 2x & ' 0 . 2x & ' 0 !. 2x 2& ' 0 132. #roblem$
127. #roblem$ The e#$ation of an as&mptote of a h&per%ola is e#$al to & ' 2x *hich 5 passes thr$ 3÷ . !etermine the 2
A c$re has an e#$ation of x2 ' 16&. "ind the len)th of the s$%-normal. A. 0. B. 0.45 . 0.5 !. 0.33
e#$ation of the h&per%ola. A. 5x2 5&2 ' 12 B. 4x2 &2 ' 16 . 4x2 &2 ' ( !. 3x2 3&2 ' 10
133. #roblem$
ENGINEERING MATHEMATICS
E2 - 13
I
ANALYTIC GEOMETRY
I
A c$re has an e#$ation of x2 16&2 16x 6& 144 ' 0. "ind the e#$ation of the tan)ent at ,(-1. A. & 1 ' 0 B. & 1 ' 0 . & 2 ' 0 !. & 2 ' 0
!. ,715 and ,(.6 13. #roblem$ A c$re has an e#$ation of &2 ' (x. omp$te the e#$ation of the tan)ent at ,24. A. x & 2 ' 0 B. 2x & 3 ' 0 . x & 2 ' 0 !. x 2& 1 ' 0
134. #roblem$ A c$re has an e#$ation of x2 16&2 16x 6& 144 ' 0. "ind the e#$ation of the normal at ,(-1. A. x ' 0 B. x ( ' 0 . x 7 ' 0 !. x 12 ' 0
140. #roblem$ A para%ola has an e#$ation of x 2 ' 16&. "ind the e#$ation of tan)ent at ,(4. A. x & ' 4 B. x 3& ' 3 . x (& ' 2 !. x (& ' 4
135. #roblem$ A c$re has an e#$ation of x2 16&2 16x 6& 144 ' 0. !etermine the eccentricit& of the c$re. A. 0.7 B. 0.76 . 0.(0 . 0.34
141. #roblem$ A circle has an e#$ation of x2 &2 ' 25. "ind the e#$ation to the tan)ent to a circle hain) a slope of 1 at the 2nd #$adrant. A. & = x + 4 3
136. #roblem$ A c$re has an e#$ation of x2 16& ' 32 4x & 2. "ind the e#$ation of the tan)ent at ,40. A. 3x 4& 12 ' 0 B. 3x 5& 11 ' 0 . 3x 6& 10 ' 0 !. 3x 4& 12 ' 0
= x +5 & = x +3 & = x +5
B. &
2
.
2
!. 2 142. #roblem$ The e#$ation of a circle is x2 &2 ' 25. "ind the e#$ation of the tan)ent on the 2nd #$adrant if it has a slope of 34. A. 3x 4& 25 ' 0 B. 3x 4& 25 ' 0 . 2x 4& 25 ' 0 !. 2x 5& 16 ' 0
137. #roblem$ A c$re has an e#$ation of & ' 2x2 1. omp$te the e#$ation of the tan)ent at ,13. A. 4x & ' 1 B. 4x & ' 1 . 4x 2& ' 1 !. 4x 3& ' 1
143. #roblem$ A c$re has an e#$ation of x2 25&2 ' 225. "ind the e#$ation of the tan)ent to the c$re at the 2 nd #$adrant. A. & = 3x + 10
13(. #roblem$ A para%ola has an e#$ation of x 2 ' 6& 10. omp$te the points of tan)enc& at the c$re. A. ,1113 and ,6( B. ,1(6 and ,47 . ,1015 and ,41
100
.
10
!.
ENGINEERING MATHEMATICS
= 3x + & = 2x − & = 2x +
B. &
10
E2 - 14
I
ANALYTIC GEOMETRY
I x2 ' 6& 10. omp$te the point of tan)enc& of the c$re. A. ,42 B. ,31 . ,41 ! ,52
144. #roblem$ A para%ola has an e#$ation of &2 ' (x. "ind the e#$ation of the diameter of the para%ola *hich %isects chords parallel to the line x & ' 4. A. & 5 ' 0 B. & 3 ' 0 . & 4 ' 0 !. & 2 ' 0
150. #roblem$ The coordinate axes are the as&mptotes of the e#$ilateral h&per%ola *hose ertex in the first #$adrant is 3 2 $nits from the ori)in. ;hat is the e#$ation of the h&per%ola. A. x& ' 4 B. x& ' . x& ' 6 !. x& ' 3
145. #roblem$ A para%ola has an e#$ation of &2 ' (x. "ind the e#$ation of the tan)ent to the para%ola hain) a slope parallel to the line x & ' 4. A. x & 2 ' 0 B. x & 2 ' 0 . x & 3 ' 0 !. x & 4 ' 0
151. #roblem$ A para%ola has an e#$ation of x2 ' - 6&. f the e#$ation of the diameter of the para%ola is x ' 3 find the slope of the chords *hich are %isected %& the diameter of para%ola. A. & ' 1 B. & ' -1 . & ' 2 !. & ' -2 152. #roblem$
146. #roblem$ A para%ola has an e#$ation of &2 ' (x. "ind the point of tan)enc&. A. ,34 B. ,24 . ,14 !. ,21
147. #roblem$ A para%ola has an e#$ation of x2 ' 6& 10. !etermine the e#$ation of a diameter of the para%ola *hich %isects chords hain) a slope of 43. A. x 2 ' 0 B. x 4 ' 0 . x 2 ' 0 !. x 4 ' 0
A c$re has an e#$ation e#$al to x2 25&2 ' 225. omp$te the second eccentricit& of the c$re. A. 2.12 B. 2.66 . 1.33 !. 1.56 153. #roblem$ A h&per%ola has an e#$ation of 2x 2 5&2 ' 10. f the e#$ation of the diameter of the h&per%ola is e#$al to x 5& ' 0 find the slope of the chords *hich are %isected %& the diameter of h&per%ola. A. & ' 2 B. & ' 1 . & ' ? !. & ' 3
14(. #roblem$ A para%ola has an e#$ation of x2 ' 6& 10. omp$te the e#$ation of a tan)ent to the c$re x 2 ' 6& 10 *hich has a slope of 43. A. 4x 3& ' 13 B. 3x 3& ' 12 . 2x 4& ' 11 !. 4x 3& ' 13 14. #roblem$ A para%ola has an e#$ation of
154. #roblem$
ENGINEERING MATHEMATICS
E2 - 15
I
ANALYTIC GEOMETRY
I
A h&per%ola has an e#$ation x& ' 16. f the e#$ation of the diameter of the h&per%ola is 3x & ' 0 find the slope of the chords *hich are %isected %& the diameter of the h&per%ola. A. & ' 3 B. & ' 2 . & ' 1 !. & ' -3
"ind the an)le of rotation of the c$re 3x& & 2 ' 0 s$ch that the transformed e#$ation *ill hae no x& term. A. 40= B. 45= . 60= !. 22.5= 160. #roblem$ A c$re has an e#$ation of r 2 sin 2C ' 6. Transform it into rectan)$lar coordinates. A. x& ' 3 B. x& ' 4 . x& ' 6 !. x& ' 2
155. #roblem$ The e#$ation of ellipse is )ien as 16x2 36&2 ' 576. omp$te the e#$ation of polar of the point ,4-6 *ith respect to the ellipse 16x 2 36&2 ' 576. A. (x 27& ' 32 B. (x 27& ' 42 . (x 27& ' 36 !. (x 27& ' 72
161. #roblem$ A c$re has an e#$ation of r 2 sin 2C ' 6. "ind the e#$ation of the diameter of the h&per%ola *hich %isects all chords hain) a slope of -2. A. 3x ' 2& B. x ' 2& . 2x ' & !. 3x ' & 162. #roblem$
156. #roblem$ The e#$ation of ellipse is )ien as 16x2 36&2 ' 576. omp$te the e#$ation of the diameter of ellipse *hich %isects all chords hain) a slope of 3. A. 4x 21& ' 0 B. 4x 27& ' 0 . 4x 27& ' 0 !. 4x 21& ' 0
A c$re has an e#$ation of r 2 sin 2C ' 6. "ind the e#$ation of the conj$)ate diameter of a h&per%ola. A. 2x & ' 0 B. 2x & ' 0 . 2x 2& ' 2 !. 2x & ' 2
157. #roblem$ The e#$ation of ellipse is )ien as 16x2 36&2 ' 576. omp$te the second eccentricit& of the ellipse. A. 1.1143 B. 1.1175 . 1.1632 !. 1.6432
163. #roblem$ A conic section is descri%e %& the follo*in) e#$ation r sin 2 C ' cos C *ith an)le C corresponds to a ri)ht trian)le *ith adjacent side x and opposite side & and h&poten$se r. omp$te the len)th of the lat$s rect$m. A. 2 B. 1 . 3 !. 0.5
15(. #roblem$ omp$te the transform e#$ation of the c$re x& ' 1 %& rotatin) the axes thro$)h 45=. A. ,x2 ,&2 ' 2 B. x & ' 2 . ,x2 ,& 2 ' 2 !. x & ' 2
164. #roblem$ A conic section is descri%ed %& the follo*in) e#$ation r sin 2 C ' cos C
15. #roblem$ ENGINEERING MATHEMATICS
E2 - 16
I
ANALYTIC GEOMETRY
I
*ith an)le C corresponds to a ri)ht trian)le *ith adjacent side x and opposite side & and h&poten$se r. omp$te the a%scissa of the foc$s of the conic section. A. 0.33 B. 0.5 . 0.25 !. 0.(
16. #roblem$ The e#$ation of a conic is r sin 2 C ' ( cos C. omp$te the coordinates of the foc$s of the )ien conic. A. ,25 B. ,20 . ,35 !. ,12 170. #roblem$
165. #roblem$ A conic section is descri%e %& the follo*in) e#$ation r sin 2 C ' cos C *ith an)le C corresponds to a ri)ht trian)le *ith adjacent side x and opposite side & and h&poten$se r. omp$te the area %o$nded %& this c$re and the lat$s rect$m. A. 0.167 B. 0.235 . 0.123 !. 0.46(
The e#$ation of a conic is r sin 2 C ' ( cos C. omp$te the area %o$nded %i the conic and the line x ' 4. A. 10.22 B. 10.33 . 10.45 !. 10.67
166. #roblem$ The polar e#$ation of a c$re is e#$al to r 2,4sin2 C cos 2 C ' 36. omp$te the area %o$nded %& the c$re. A. 1(.44 B. 1(.(5 . 12.23 !. 17.42
171. #roblem$ The polar e#$ation of the c$re is expressed as
r =
2 1 − sin θ
. omp$te
the ordinate to the ertex of the c$re. A. -1 B. 2 . -3 !. 1
167. #roblem$ The polar e#$ation of a c$re is e#$al to r 2,4sin2 C cos 2 C ' 36. omp$te the total len)th of the c$re. A. 12.6( B. 12.33 . 14.67 !. 16.02
172. #roblem$ +oint F+G hain) a c&lindrical coordinates of ,(30=5. "ind the al$e of x in artesian coordinates. A. 7.42 B. 10.34 . 13.66 !. 6.3
16(. #roblem$ The polar e#$ation of a c$re is e#$al to r 2,4sin2 C cos 2 C ' 36. !etermine the eccentricit& of the )ien c$re. A. 0.(45 B. 0.334 . 0.745 !. 0.232
173. #roblem$ +oint F+G hain) a c&lindrical coordinates of ,(30=5. "ind the al$e of & in artesian coordinates. A. 5 B. 6 . 4 !. 2
ENGINEERING MATHEMATICS
E2 - 17
I
ANALYTIC GEOMETRY
I .
174. #roblem$ +oint F+G hain) a rectan)$lar coordinates of ,345. "ind the al$e of H in +olar coordinates. A. 64=54 B. 54=54 . 46=45 !. 54=46
5 2
!. 5 3 1(0. #roblem$ +oint FG hain) a artesian coordinates of ,345. "ind the al$e of C $sin) spherical coordinates. A. 53=0( B. 45=34 . 76=12 !. 45=
175. #roblem$ +oint F+G hain) a rectan)$lar coordinates of ,345. "ind the al$e of I in polar coordinates. A. 52=21 B. 57=67 . 55=54 !. 55=33
1(1. #roblem$ +oint FG hain) a artesian coordinates of ,345. "ind the al$e of I $sin) spherical coordinates. A. 53=0( B. 45=34 . 76=12 !. 45=
176. #roblem$ +oint FAG hain) a artesian coordinates of ,345. "ind the al$e of r $sin) c&lindrical coordinates. A. 4 B. 2 . 6 !. 5
1(2. #roblem$ A plane has an e#$ation of 4x & (8 33 ' 0. "ind the distance %et*een the point A,15-3 from the plane. A. 2 B. 3 . 4 !. 5
177. #roblem$ +oint FAG hain) a artesian coordinates of ,345. "ind the al$e of C $sin) c&lindrical coordinates. A. 57=46 B. 53=0( . 33=55 !. 5=42
1(3. #roblem$ A plane has an e#$ation of 4x & (8 33 ' 0. "ind the an)le %et*een the planes 4x & (8 33 ' 0 and 2x 3& 8 2 ' 0. A. 67=17 B. 55=44 . 33=23 !. 6=56
17(. #roblem$ +oint FAG hain) a artesian coordinates of ,345. "ind the al$e of 8 $sin) c&lindrical coordinates. A. 5 B. 3 . 2 !. 1
1(4. #roblem$ "ind the distance %et*een points A,515 and B,432. A. 34
17. #roblem$ +oint FG hain) a artesian coordinates of ,345. "ind the al$e of r $sin) spherical coordinates. A. 4 3
B.
14
.
56
!.
67
1(5. #roblem$
B. 2 5
ENGINEERING MATHEMATICS
E2 - 18
I
ANALYTIC GEOMETRY
I
The distance from A,1(3 to B,x24 is e#$al to 7.2( find the al$e of x. A. 2 or 3 B. 2 or -5 . 5 or -3 !. 5 or -2
A plane is descri%ed %& the e#$ation 2x & 38 ' 12. ;hat is the coordinate of the point on the plane *hich is closest to the &-axis. A. -14 B. -12 . -23 !. -34
1(6. #roblem$ "ind the an)le %et*een t*o lines *hose direction parameters are ,-122 and ,4-1(. A. 56=16 B. 76=14 . 6(=16 !. 22=12
11. #roblem$ A plane is descri%ed %& the e#$ation 2x & 38 ' 12. ;hat is the smallest an)le that the )ien plane maes *ith the plane 2x 3& 48 ' . A .15=7( B. 12=(( . 1=45 !. 32=56
1(7. #roblem$ The e#$ation of the plane passin) thr$ points ,541 ,4-2-3 and ,065 is expressed as x & 8 + + = 1 . "ind the al$e of B A B A. 13 B. 23 . -13 !. -23
12. #roblem$ The ertices of a trian)le are A,110 B,101 and ,011. "ind the distance AB. A. 3
1((. #roblem$ The e#$ation of the plane passin) thr$ points ,541 ,4-2-3 and ,065 is expressed as x & 8 + + = 1 . "ind the e#$ation of A B the plane. A. 2x 3& 48 ' 2 B. 2x & 48 ' 2 . 2x 3& 48 ' 2 !. 2x & 48 ' 2
B.
6
.
2
!.
5
13. #roblem$ The ertices of a trian)le are A,110 B,101 and ,011. "ind the points of intersection of the medians of the trian)le. 2 2 2 A. ÷ 3 3 3 B.
1(. #roblem$ A plane is descri%ed %& the e#$ation 2x & 38 ' 12. "ind the distance from the point ,146 to the )ien plane. A. 1.26 B. 1.054 . 1.06 !. 1.035
. !.
3 3 3 2 2 2÷ 5 3 1 2 2 2÷ 2 4 5 3 3 3÷
14. #roblem$ The points ,126 ,162 are ertices of an e#$ilateral trian)le. f the x and & coordinates of the other ertex is 5 and 2. "ind the al$e of 8.
10. #roblem$
ENGINEERING MATHEMATICS
E2 - 19
I
ANALYTIC GEOMETRY
I
A. 5 B. 3 . 2 !. 1
"ind the ne* coordinates of the point ,33 if the axes is translated to the ne* ori)in at ,-24. A. ,51 B. ,41 . ,5-2 !. ,5-1
15. #roblem$ The first three consec$tie ertices of a parallelo)ram are ,64-1,767 and ,17-5. f x ' 0 and & ' 5 of the fo$rth ertex find the al$e of 8. A. 2 B. 6 . !. 1
201. #roblem$ omp$te the ori)inal coordinates *ith respect to x and & axes if the translated coordinates of this point at a ne* ori)in ,32 is e#$al to ,4-3. A. ,46 B. ,7-1 . ,4-2 !. ,5-1
16. #roblem$ "ind the ne* e#$ation of the line 5x 4& 3 ' 0. if the ori)in is translated to the point ,12. A. 5x 4& 16 ' 0 B. 4x 3& 12 ' 0 . 3x 4& 10 ' 0 !. 4x 4& 12 ' 0
202. #roblem$ ;hat conic section is represented %& x2 4&2 (x 4& ' 15< A. h&per%ola B. circle . ellipse !. para%ola
17. #roblem$ "ind the ne* e#$ation of the c$re x2 4x 7& ' 0 if the ori)in is translated to point ,-26. A. ,x2 7& 3( ' 0 B. ,x2 (& 32 ' 0 . ,x2 6& 2( ' 0 !. ,x2 12& 24 ' 0
203. #roblem$ f the )eneral e#$ation of the conic is Ax2 2Bx& &2 2!x 2:& " ' 0 and B2 A J 0 then the conic isK A. circle B. para%ola . ellipse !. h&per%ola
1(. #roblem$ B& translation of axes simplif& the e#$ation x2 6x 6& 15 ' 0. A. ,x2 ' 6& B. ,x2 ' 5& . ,x2 ' 7& !. ,x2 ' 6&
204. #roblem$ ;hat t&pe of conic has an e#$ation of Ax2 &2 !x :& " ' 0. A. circle B. para%ola . ellipse !. h&per%ola
1. #roblem$ "ind the ne* coordinates of the point ,3-5. f the axes are translated to the ne* ori)in at ,-46. A. ,7-11 B. ,610 . ,-105 !. ,4-16
205. #roblem$ 3x2 2x 5& 7 ' 0 determine the c$re. A. circle B. ellipse . h&per%ola !. para%ola
200. #roblem$
ENGINEERING MATHEMATICS
E2 - 20
I
ANALYTIC GEOMETRY
I
206. #roblem$ 212. #roblem$ f the ertices of a trian)le are A,000 B,246 and ,24-4. "ind the point of intersection of the median of the trian)le. 3 4 6 A. ÷ 4 3 5
The e#$ation x 2 &2 4x 2& 20 ' 0 descri%es K A. circle B. ellipse . para%ola !. h&per%ola 207. #roblem$ ;hat conic section is represented %& x2 16&2 36x 32& 2 ' 0. A. circle B. para%ola . ellipse !. h&per%ola
B. . !.
3 3 2 4 ( 3÷ 4 ( 2 3 3 3÷ 6 ( 4 5 3 3÷
213. #roblem$ f the ertices of a trian)le are A,000 B,246 and ,24-4. "ind the len)th of the shortest median of the trian)le. A. 4.67 B. 4.0 . 4.5( !. 3.67
20(. #roblem$ ;hat conic section is 2x2 (x& 4x ' 12< A. h&per%ola B. ellipse . para%ola !. circle 20. #roblem$ ;hat conic section is defined %& the e#$ation 7x2 16&2 16x& 60x 6& 156 ' 0. A. h&per%ola B. ellipse . para%ola !. circle
214. #roblem$ f the ertices of a trian)le are A,000 B,246 and ,24-4. "ind the len)th of the lon)est median of the trian)le. A. .67 B. (.31 . 5.42 !. 10.7
210. #roblem$ A point has a coordinate of ,23-6. "ind the distance from the ori)in to the point. A. 4 B. 6 . 7 !.
215. #roblem$ A *areho$se roof needs a rectan)$lar s&li)ht *ith ertices ,300 ,330 ,004 ,034. f the $nits are in meters find the area of the s&li)ht. A. 25 s#. m. B. 22.5 s#. m. . 15 s#. m. !. 12.5 s#. m.
211. #roblem$ "ind the direction cosines on the xaxis of the line from ,-124 to ,105. A. 43 B. 23 . 32 !. 52
216. #roblem$ ;hat is the radi$s of a sphere *hose center is at the ori)in that passes thro$)h ,(16< A. 101
ENGINEERING MATHEMATICS
E2 - 21
I
ANALYTIC GEOMETRY
B.
.
105
!.
7
I "ind the e#$ation of a sphere of radi$s 3 and tan)ent to all three coordinate planes if the center is in the first octant. A. x2 &2 82 6x 6& 68 1( '0 B. x2 &2 82 6x 6& 68 1( '0 . x2 &2 82 6x & 68 1( '0 !. x2 &2 82 6x 6& 68 1( '0
217. #roblem$ ;hat is the e#$ation of sphere *ith radi$s 5 and center at ,-235. A. ,x h2 ,& 2 ,8 j 2 ' 25 B. ,x h2 ,& 2 ,8 j 2 ' 20 . ,x h2 ,& 2 ,8 j 2 ' 15 !. ,x h2 ,& 2 ,8 j 2 ' 10
223. #roblem$ "ind the ol$me of the solid hain) coordinates of ,004 ,300 and ,000. A. 16 c$.$nits B. 14 c$.$nits . 12 c$.$nits !. ( c$.$nits
21(. #roblem$ A sphere has an e#$ation of x2 &2 82 2x (& 168 65 ' 0. "ind the centroid of the sphere. A. ,16-( B. ,1-4-( . ,471 !. ,5-1
224. #roblem$ "ind the component of the line se)ment from the ori)in to the point ,63 on a line *hose direction
21. #roblem$ A sphere has an e#$ation of x2 &2 82 2x (& 168 65 ' 0. "ind the s$rface area of the sphere. A. 74L s#.$nits B. 42L s#.$nits . 64L s#.$nits !. 4L s#.$nits
cosines are
2 1 2 - - . 3 3 3
A. 15 B. 17 . 14 !. 11
220. #roblem$ A )ien sphere has an e#$ation of x2 &2 82 4x 6& 108 13 ' 0. omp$te the centroid of the sphere. A. ,-235 B. ,-647 ,24( !. ,-234
225. #roblem$ "ind the len)th of the radi$s ector of ,213. A. 5.(5 B. 7.33 . 3.74 !. 3.11
221. #roblem$ A )ien sphere has an e#$ation of x2 &2 82 4x 6& 108 13 ' 0. "ind the ol$me of the sphere. A. 346.7 c$.$nits B. 523.6 c$.$nits . 633.5 c$.$nits !. 445.7 c$.$nits
226. #roblem$ A line has an e#$ation of x 5& 5 ' 0. "ind the e#$ation of the line thro$)h point ,31 that is parallel to this line. A. x 5& ( ' 0 B. 5x & 14 ' 0 . 2x 3& ' 3 !. x 5& ( ' 0
222. #roblem$ ENGINEERING MATHEMATICS
E2 - 22
I
ANALYTIC GEOMETRY
I
227. #roblem$ A line has an e#$ation of x 5& 5 ' 0. "ind the e#$ation of the line thro$)h point ,31 that maes an an)le of 45= cloc*ise from the line that is perpendic$lar to the line x 5& 5 ' 0 at that point. A. x 5& ( ' 0 B. 5x & 14 ' 0 . 2x 3& ' 3 !. x 5& ( ' 0
!. 2x 3& ' 0
232. #roblem$ A line se)ment has its ends on the coordinate axes and forms *ith them a trian)le of area e#$al to 36 s#. $nits The se)ment passes thro$)h the point ,52. omp$te the len)th of the line se)ment intercepted %& the coordinate axes. A. 16.34 B. 17.4 . 13.42 !. 12.66
22(. #roblem$ A line connectin) coordinates ,x7 and ,10& is %isected at ,(2. "ind the al$e of x. A. 4 B. 5 . 6 !. 7 22. #roblem$ A line connectin) coordinates ,x7 and ,10& is %isected at ,(2. "ind the e#$ation of the line. A. 5x 2& 44 ' 0 B. 2x 5& 44 ' 0 . 4x 2& 44 ' 0 !. 3x 5& 22 ' 0
233. #roblem$ A line se)ment passes thro$)h point ,22. f the len)th of the line se)ment intercepted %& the coordinate axes is e#$al to the s#$are root of 5. omp$te the e#$ation of the line. A. 2x & 2 ' 0 B. 3x & 2 ' 0 . 2x & 2 ' 0 !. 3x & 2 ' 0
230. #roblem$ A line se)ment has its ends on the coordinate axes and forms *ith them a trian)le of area e#$al to 36 s#. $nits The se)ment passes thro$)h the point ,52. ;hat is the slope of the line se)ment. A. 2 B. -2 . 3 !. -3
234. #roblem$ A line se)ment passes thro$)h point ,22. f the len)th of the line se)ment intercepted %& the coordinate axes is e#$al to the s#$are root of 5. omp$te the area of the trian)le *hich forms *ith the coordinate axes. A. 2 s#. $nits B. 1 s#. $nits . 4 s#. $nits !. 0.5 s#. $nits
231. #roblem$ A line se)ment has its ends on the coordinate axes and forms *ith them a trian)le of area e#$al to 36 s#. $nits The se)ment passes thro$)h the point ,52. omp$te the e#$ation of the line se)ment. A. 2x 3& ' 12 B. 2x & ' 12 . x 2& ' 0
235. #roblem$ The s$m of the coefficients of x and & in Ax B& 16 ' 0 is 14. f the
ENGINEERING MATHEMATICS
E2 - 23
I
ANALYTIC GEOMETRY
I
slope of the line is (. "ind the al$e of B. A. 3 B. 1 . 2 !. -2
. 17x 45& 67 ' 0 !. 45x 17& 67 ' 0
236. #roblem$ The line 3x 2& 10 ' 0 is perpendic$lar to 2x B& 2 ' 0. "ind the distance from the intersection of the lines to the ori)in. A. 3.( B. 2.(2( . 5.235 !. 6.12
240. #roblem$ n trian)le AB A,x4 is e#$idistant from B,5-2 and ,34. "ind the al$e of x. A. 10 B. 11 . 12 !. 13
237. #roblem$ The ertices of a trian)le are at A,12 B,3( and ,(-1. ocate the point of intersection of its medians ,centroid of A. A. centroid is at ,43 B. centroid is at ,34 . centroid is at ,53 !. centroid is at ,35
241. #roblem$ A para%ola has its foc$s at ,7-4 and its directrix has an e#$ation e#$al to & 2 ' 0. "ind the ertex of the para%ola. A. ,45 B. ,7-1 . ,63 !. ,75
23(. #roblem$ The ertices of a trian)le are at A,12 B,3( and ,(-1. ocate the point of intersection of its altit$des ,orthocenter. 1 1 A. ÷ 4 12 B. . !.
242. #roblem$ A para%ola has its foc$s at ,7-4 and its directrix has an e#$ation e#$al to & 2 ' 0. omp$te the len)th of lat$s rect$m. A. 15 B. 13 . 12 !. 1(
3 12 4 1÷ 4 12 3 ÷ 1 17 ( 12÷
243. #roblem$ A para%ola has its foc$s at ,7-4 and its directrix has an e#$ation e#$al to & 2 ' 0. omp$te the e#$ation of para%ola. A. x2 14x 12& 61 ' 0 B. x2 14x 6& 32 ' 0 . x2 15x 5& 4 ' 0 !. x2 15x 25& 4 ' 0
23. #roblem$ The ertices of a trian)le are at A,12 B,3( and ,(-1. f the :$lers line of a trian)le passes thro$)h the centroid and orthocenter of the trian)le find the e#$ation of the :$lers line of this trian)le. A. 67x 45& 17 ' 0 B. 45x 17& 67 ' 0
244. #roblem$ A para%ola has its axis parallel to the &-axis one end of its lat$s rect$m is at ,6 and the ertex is at ,54. "ind the len)th of the lat$s rect$m.
ENGINEERING MATHEMATICS
E2 - 24
I
ANALYTIC GEOMETRY
I
A. 7 B. ( . 2 !. 5
t*o fixed points ,03 and ,0-3 is al*a&s e#$al to (. omp$te the e#$ation of ellipse. A. 16x2 7&2 ' 112 B. 16x2 7&2 ' 112 . 7x2 16&2 ' 112 !. 7x2 16&2 ' 112 250. #roblem$ An ellipse is a loc$s of a point so that the s$m of its distances from the t*o fixed points ,03 and ,0-3 is al*a&s e#$al to (. omp$te the distance %et*een their directrices. A. 16.64 B. 11.(5 . .44 !. 10.66
245. #roblem$ A para%ola has its axis parallel to the &-axis one end of its lat$s rect$m is at ,6 and the ertex is at ,54. "ind the e#$ation of the para%ola. A. x2 10x (& 57 ' 0 B. x2 10x 16& 57 ' 0 . x2 10x (& 57 ' 0 !. x2 10x (& 57 ' 0 246. #roblem$ The distance from point A
(
)
6 cos 45° 2 sin 45° to the center
251. #roblem$ The len)th of the conj$)ate axis of an e#$ilateral h&per%ola is e#$al to 12. The x and &-axis forms the as&mptotes of the e#$ilateral h&per%ola. ;hat is the distance from the ertex to the intersection of the x and & axis. A. 7 B. 6 . 3 !. 13
of ellipse is e#$al to 2. f the e#$ation of ellipse 2x 2 6&2 ' 12. omp$te the len)th of the lon)er focal radi$s from point A. A. 4.42 B. 3.(6 . 7.36 !. (.13 247. #roblem$ The loc$s of a point *hich moes so that the s$m of its distances from t*o fixed points ,30 and ,-30 is al*a&s e#$al to 10. omp$te the len)th of the minor axis. A. ( B. 12 . 7 !. 14
252. #roblem$ The len)th of the conj$)ate axis of an e#$ilateral h&per%ola is e#$al to 12. The x and &-axis forms the as&mptotes of the e#$ilateral h&per%ola. ;hat is the e#$ation of the e#$ilateral h&per%ola< A. x& ' 36 B. x& ' 14 . x& ' 12 !. x& ' 1(
24(. #roblem$ The loc$s of a point *hich moes so that the s$m of its distances from t*o fixed points ,30 and ,-30 is al*a&s e#$al to 10. omp$te the eccentricit& of the ellipse. A. 0.30 B. 0.60 . 0.76 !. 0.6(
253. #roblem$ A h&per%ola *hose transerse axis is parallel to the x-axis *ith its center at ,00 has an as&mptote *ith a slope of 0.4 ertical to 1 hori8ontal. The distance %et*een the ertices is e#$al to 10. omp$te the e#$ation of the h&per%ola. A. 5x2 26&2 ' 100 B. 4x2 25&2 ' 100
24. #roblem$ An ellipse is a loc$s of a point so that the s$m of its distances from the ENGINEERING MATHEMATICS
E2 - 25
I
ANALYTIC GEOMETRY
I
. 3x2 16&2 ' 100 !. 6x2 16&2 ' 100 254. #roblem$ A h&per%ola *hose transerse axis is parallel to the x-axis *ith its center at ,00 has an as&mptote *ith a slope of 0.4 ertical to 1 hori8ontal. The distance %et*een the ertices is e#$al to 10. omp$te the e#$ation of the as&mptote. A. 2x 5& ' 0 B. 3x 7& ' 0 . 3x 4& ' 0 !. 4x (& ' 0 255. #roblem$ A h&per%ola *hose transerse axis is parallel to the x-axis *ith its center at ,00 has an as&mptote *ith a slope of 0.4 ertical to 1 hori8ontal. The distance %et*een the ertices is e#$al to 10. omp$te the eccentricit& of the h&per%ola. A. 1.604 B. 1.56 . 1.07( !. 1.334
ENGINEERING MATHEMATICS
E2 - 26
I
