MATHEMATICS ANALYTIC GEOMETRY CE LICENSURE EXAMINATION PROBLEMS
LINES
ANALYTIC GEOMETRY .
4!at is t!e e8#ation e8#ation of t!e line t!at ,asses t!ro#g! t!ro#g! (4, 0) and 0) and is ,arallel to t!e line x line x – y – 2 = 0. 0. $M70 M /2' 9:":*;% 9<"<*;% a. c. 9<":*;% 9:"<*;% b. d.
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A line ,assing t!ro#g! (5, 3) !as 3) !as a slo,e of 3 / 2. ind t!e distance+eas#red along t!e line- fro+ t!e ,oint (5, 3) to 3) to t!e Y (a"is. (a"is. $M%/ M /&' ) 2 a. c. 0.2*/ 7.%&* b. d.
CENTROID OF PLANE AREAS 1.
2.
3.
Two vertices of a triangle are (6, -1) and -1) and (7, -3). -3). ind t!e ordinate of t!e t!ird verte" s#c! t!at t!e centroid of t!e triangle will lie at t!e origin. $M%% M &%' (&) &) a. c. * (* b. d. A co+,osite area consist of a rectangle wit! vertices at coordinates (3, 0)0)- (9, 0)0)- (9, 3)3)- and and (3, 3) 3) and a triangle wit! vertices at coordinates (3, 3)3)- (9, 3)3)- and (7, 5). 5). How far fro+ t!e Y (a"is (a"is is t!e centroid of t!e co+,osite area $M%/ M /%' 0.%1) 0./2* a. c. 2.3*& 0.13& b. d. A triangle !ave vertices at A at A (-3, -2)-2)- B (2, 6)6)- and C (4, 2). 2). 4!at is t!e a5scissa of t!e centroid of t!e triangle $N%/ M &&' ) ) 6* 6/ a. c. 2 6* & b. d.
ANGLE OF INCLINATION OF LINES ".
DISTANCE BETWEEN A POINT AND A LINE #.
T!e distance fro+ t!e ,oint (2, 1) to 1) to t!e line 4x – 3y + 5 = 0 is $M7* M &/' / & a. c. * ) b. d.
1$.
T!e two ,oints on t!e lines 2x + 3y + 4 = 0 w!ic! 0 w!ic! are at distance 2 fro+ t!e line 3x + 4y – 6 = 0 are> $M7* M &*' $(1- (1' and $(&0- (&0' $(2.2- &' and $(2- /' a. c. $(**- 0*' and $(2- /' $0*- (**' and $*- (*' b. d.
11.
?eter+ine t!e distance fro+ (5, 10) to 10) to t!e line x line x – y = 0. $N70 M &0' ).10 ).2* a. c. ).01 ).3/ b. d.
AREA BY COORDINATES 4.
ind t!e area of triangle w!ose vertices are A (-3, -1)-1) - B (5, 3) and 3) and C (2, -8). -8). $N70 M 0' )* )0 a. c. )1 )/ b. d.
DISTANCE BETWEEN TWO POINTS 5.
ind t!e angle t!at t!e line 2y – 9x – 18 = 0 +a=es 0 +a=es wit! t!e X t!e X (a"is. (a"is. $M77 M *' 3*.33° *3.33° a. c. *.2° 33.*3° b. d.
ind t!e distance 5etween (4, -2) and -2) and (-5, 1). 1). $M73 M )%' 7.03 7.*7 a. c. 7.12 7.7* b. d.
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MATHEMATICS ANALYTIC GEOMETRY 12.
ind t!e distance fro+ t!e ,oint (5, -3) to t!e line 7x – 4y – 28 = 0 . $N71 M //' /.0/ /.*1 a. c. /.)0 /.2* b. d.
2$.
?eter+ine t!e area enclosed 59 t!e c#rve x2 – 10x + 4y + y 2 = 196 . $M77 M &0' &2π &/π a. c. //2π &**π b. d.
13.
How far is t!e line 3x – 4y + 15 = 0 fro+ t!e origin $N77 M &/' & ) a. c. / * b. d.
21.
ind t!e center of t!e circle x2 + y2 – 6x + 4y – 23 = 0 . $N77 M &)' $)- (/' $()- /' a. c. $)- /' $()- (/' b. d.
14.
How far is t!e line 5x – 4y + 7 = 0 fro+ t!e ,oint (-2, 8) $M%& M 7' *.32 )./& a. c. 7.1* 2.*3 b. d.
22.
ind t!e area of t!e c#rve x2 + y2 + 6x – 12y + 9 = 0. $N%% M )' &/2 s8#are #nits 7/ s8#are #nits a. c. &&) s8#are #nits &)1 s8#are #nits b. d.
23.
4!at is t!e radi#s of t!e circle x2 + y 2 – 10x + 4y – 196 = 0 $N7* M &&' &2 &3 a. c. &0 &* b. d.
?eter+ine t!e area enclosed 59 t!e c#rve x2 + y 2 + 8x + 4y – 61 = 0. $M%& M )%' )*&./) /&3.02 a. c. *)/.13 /2*.*3 b. d.
24.
4!at is t!e radi#s of t!e circle x2 + y2 – 6y = 0 $M72 M 2' ) 0 a. c. 2 * b. d.
Locate t!e center of t!e c#rve x2 + y 2 + 8x + 4y – 61 = 0 . $N%& M )' $*- (/' $*- /' a. c. $(*- /' $(*- (/' b. d.
25.
?eter+ine t!e distance fro+ t!e c#rve x2 + y 2 – 16y + 16x + 64 = 0 to ,oint (6, 4). $M%/ M 3' 0.20 3.2/ a. c. 2./* 1.03 b. d.
2.
4!at is t!e radi#s of t!e circle x2 + y 2 + 4x – 10y – 92 = 0 $M%) M &/' &/ && a. c. &) &% b. d.
CIRCLES 15.
1.
1!.
1".
1#.
4!ere is t!e center of t!e c#rve x2 + y2 – 2x – 4y – 31 = 0 $N72 M &&' a. $(&- (/' c. $&- /' $&- (/' $(&- /' b. d. How far fro+ t!e Y (a"is is t!e center of t!e c#rve 2x2 + 2y 2 + 10x – 6y – 55 = 0 $M70 M /)' ().% ()./2 a. c. (/.32 (/.2 b. d. 2
2
?eter+ine t!e lengt! of t!e c!ord co++on to t!e circles x + y = 64 and x2 + y2 – 16x = 0. $N71 M 1' &).10 &)./2 a. c. &/.1/ &/./1 b. d.
PARABOLA 2!.
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4!ere is t!e verte" of t!e ,ara5ola x2 = 4 (y – 2) $N7* M &%' $/- %' $%- *' a. c. $*- %' $%- /' b. d.
MATHEMATICS ANALYTIC GEOMETRY 2".
4!at is t!e lengt! of t!e lat#s rect#+ of t!e c#rve x2 = 20y $M72 M *' 2 * a. c. /% 1 b. d.
35.
ind t!e ratio of t!e +a@or a"is to t!e +inor a"is of t!e elli,se 9x2 + 4y2 – 72x – 24y – 144 = 0 . $M77 M 2' %.03 &.2 a. c. &.1 %.32 b. d.
2#.
4!at is t!e lengt! of t!e lat#s rect#+ of t!e c#rve x2 = -12y $M71 M 2' &/ ) a. c. () (&/ b. d.
3.
T!e area of t!e elli,se is given as A = 3.1416ab. ind t!e area of t!e elli,se 25x2 + 16y 2 – 100x + 32y = 284 . $N77 M 1' 10./ s8#are #nits 01./ s8#are #nits a. c. 0/.1 s8#are #nits 1/.0 s8#are #nits b. d.
3$.
?eter+ine t!e e8#ation of t!e directri" of t!e c#rve x2 = 16y. $M%% M /7' ":* ; % 9<* ; % a. c. "<* ; % 9:* ; % b. d.
3!.
ind t!e distance 5etween t!e foci of t!e c#rve 9x2 + 25y2 – 18x + 100y – 116 = 0. $N%% M *' 3 1 a. c. 0 &/ b. d.
3".
?eter+ine t!e lat#s rect#+ of t!e elli,se 49x2 + 36y2 + 392x – 216y – 656 = 0. $N%& M *' &0.))) &)./&2 a. c. &%./10 7.*20 b. d.
3#.
T!e distance fro+ t!e foc#s to t!e vertices of an elli,se are * and 0 #nits. ?eter+ine t!e second eccentricit9 of t!e elli,se. $N%& M /1' %./%* %.2%) a. c. %.)%2 %.*%/ b. d.
4$.
4!at is t!e distance 5etween t!e vertices of t!e elli,se 64x2 + 25y 2 + 16x – 16y – 648 = 0 $M%/ M //' 0.)/* &%./& a. c. b. &/.2* d. 2.&%2
41.
ind t!e ratio of t!e lengt! of t!e +inor a"is to t!e lengt! of t!e +a@or a"is of t!e elli,se 9x2 + 16y 2 – 144 = 0. $M%) M &&' %.32 %.*) a. c. %.0/ %.21 b. d.
42.
If t!e c#rve Ax2 + By 2 + F = 0 ,asses t!ro#g! (0, 3) and (4, 0)- t!e c#rve is> $N%) M &/' a ,ara5ola a circle a. c. an elli,se a !9,er5ola b. d.
ELLIPSE 31.
32.
33.
34.
4!at is t!e area enclosed 59 t!e c#rve 9x2 + 25y 2 – 225 = 0 $N7* M &/' &11.*70 &2%.370 a. c. *3.&/2 32.)71 b. d. Locate t!e center of t!e c#rve x2 + 16y2 – 16x + 96y + 144 = 0 . $N73 M &&' $1- ()' $&*- ()' a. c. $/- ()' $1- (&' b. d. ?eter+ine t!e lengt! of t!e lat#s rect#+ of t!e c#rve 25x2 + 9y 2 – 300x – 144y + 1251 = 0 . $N73 M &)' )./ ).0 a. c. ).* ).% b. d. How far a,art are t!e directrices of t!e c#rve 25x2 + 9y2 – 300x – 144y + 1251 = 0 $M71 M /1' &*./ &/.2 a. c. &).2 &2./ b. d.
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MATHEMATICS ANALYTIC GEOMETRY %YPERBOLA 43.
How far fro+ t!e X (a"is is t!e foci of t!e !9,er5ola x2 – 2y2 + 4x + 4y + 4 = 0 $N72 M *' /.0/ /.3) a. c. /.*2 /./* b. d.
SPACE ANALYTIC GEOMETRY 44.
If t!e distance 5etween ,oints A (2, 10, 4) and B (8, 3, z) is 9.434w!at is t!e val#e of z $N%/ M &/' * ) a. c. 0 2 b. d.
45.
?eter+ine t!e radi#s of t!e s,!ere w!ose e 8#ation is x2 + y2 + z 2 + 8x – 2y + 1 = 0 . $N%/ M &)' 2 * a. c. ) / b. d.
4.
oints C (5, 7, z) and (4, 1, 6) are 7.28 !" a,art. ind t!e val#e of z . $M%) M &)' ) / a. c. * & b. d.
4!.
T!e distance 5etween ,oints (x, 3, 6) and (2, 10, 4) is 9.434 #nits. Calc#late t!e val#e of x. $N%) M &)' 2 1 a. c. b. 3 d. 0
4".
If t!e distance 5etween ,oints A (2, 5, 4) and B (x, 3, 6) is AB = 5.745 "- deter+ine t!e val#e of x. $N%) M &*' * ) a. c. 0 3 b. d.
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