MATHEMATICS
EXERCISE-1
Q.1
The equation to the parabola whose focus is (0, – 3) and directrix is y = 3 is (1) x2 = – 12y (2) x2 = 12y (3) y2 = 12x (4) y2 = – 12x
Q.2
If (0, 0) be the vertexand 3x– 4y+ 2 = 0 bethe directrix of a parabol,thenthe length ofits latus rectum is (1)
4 5
(2)
2
(3)
5
8 5
(4)
1 5
Q.3
If 2x + y + = 0 is a focal chord of the parabola y2 = – 8x, then the value of is (1) – 4 (2) 4 (3) 2 (4) – 2
Q.4
The distance between the focus and the directrix of the parabola x2 =8y,is (1) 8 (2) 2 (3) 4 (4) 6
Q.5
For any parabola focus is (2, 1)and directrix is2x – 3y + 1 = 0, then equation of the latus rectum is (1) 3x + 2y + 8 = 0 (2) 2x – 3y – 1 = 0 (3) 2x – 3y + 1 = 0 (4) 3x – 2y + 4 = 0
Q.6
The area of the triangle formed by the lines joining the vertex of the parabola x2 = 12yto theendsof its rectum is (1) 16 sq. units (2) 12 sq. units (3) 18 sq. units (4) 24 sq. units
Q.7
Vertex of the parabola 9x2 – 6x + 36y + 9 = 0 is
1 2 (1) , 3 9 Q.8
Q.10
1 1 , 3 2
(3)
1 1 (4) , 3 2
The focus of the parabola y2 – x – 2y + 2 = 0 is (1) (1, 2)
Q.9
1 1 , 3 2
(2)
1 (2) , 0 4
3 (3) , 1 4
The equation of the axis of the parabola x2 – 4x – 3y + 10 = 0, is (1) y + 2 = 0 (2) x + 2 = 0 (3) x – 2 = 0
5 (4) , 1 4
(4) y – 2 = 0
The point of intersection of the latus rectum and axes of the parabola y2 + 4 x + 2 y – 8 = 0 i s
5 (1) , 1 4
7 (2) , 5
5
2
9 (3) , 1 4
(4) none of these
Q.11
Which of the following is not the equation ofparabola (1) 4x2 + 9y2 – 12xy + x + 1 = 0 (2) 4x2 – 12xy + 9y2 + 3x + 5 = 0 (3) 2x2 + y2 – 4xy = 8 (4) 4x2 + 9y2 – 12xy + x + 1 = 0
Q.12
If vertex and focus of a parabola are on x-axis and at distance p and q respectively from the origin, then its equation is (1) y2 = – 4 (p – q) (x + p) (2) y2 = 4 (p – q) (x – p) (3) y2 = – 4 (p – q) (x – p) (4) none of these
Q.13
If (2, 0) and (5, 0) are the vertex and focus of a parabola respectively then its equation is (1) y2 = – 12x – 24 (2) y2 = 12x – 24 (3) y2 = 12x + 24 (4) y2 = – 12x + 24
Q.14
Any point on the parabola whose focus is (0, 1)and the directrix is x + 2 = 0 is given by (1) (t2 + 1, 2t + 1) (2) (t2 + 1, 2t – 1) (3) (t2, 2t) (4) (t2 – 1, 2t + 1)
Q.15
The length of the chord of the parabola y2 = 4x which passes through the vertex and makes 30° angle with x-axis is (1)
3
(2)
2
3
(3) 8 3
2
(4)
3
Q.16
The length of the intercept made by the parabola x2 – 7x + 4y + 12 = 0 on x-axis is (1) 4 (2) 3 (3) 1 (4) 2
Q.17
If PSQ is the focal chord of the parabola y2 = 8x such that SP = 6. Thenthe length SQ is (1) 4 (2) 6 (3) 3 (4) none of these
Q.18
If the line x + y – 1 = 0 touches the parabola y2 = kx,then the value of k is (1) 2 (2) – 4 (3) 4 (4) – 2
Q.19
The line y = mx+ c may touch the parabola y2 =4a(x+a),if
a m
(1) c = am – Q.20
a m
(3) c =
a m
a m
(4) c = am +
Forwhat value of k, the line 2y– x + k = 0 touches the parabola x2 + 4 y = 0 (1) 2
Q.21
(2) c =
(2)
1 2
(3) – 2
(4)
1 2
The point where the line x + y = 1 touches the parabola y = x – x2, is
1 1 (1) , 2 2
(2) (1, 0)
(3) (0, 1)
(4) (– 1, – 2)
Q.22
The equation of the tangent to the parabola y = 2 + 4x – 4x2 with slope – 4 is (1) 4x + y – 6 = 0 (2) 4x + y + 6 = 0 (3) 4x – y – 6 = 0 (4) none of these
Q.23
The point onthe curve y2 = x the tangent at which makes an angle of 45° with x-axiswill be given by
1 1 (1) , 2 2
1 1 (2) , 2 4
(3) (2, 4)
1 1 (4) , 4 2
Q.24
The equation of the common tangents to the parabolas y2 =4xandx2 =32yis (1) x + 2y = 4 (2) x = 2y + 4 (3) x = 2y – 4 (4) x + 2y + 4 = 0
Q.25
If m1 andm2 are slopes of the two tangents that are drawn from (2, 3) to the parabola y2 = 4x then value
1 1 = of m m 2 1 (1) – 3
(2) 3
(3)
2 3
(4)
3 2
Q.26
The equation of common tangent to the circle x2 + y2 = 2a2 and parabola y2 =8axis (1) y = x + a (2) y = ± x ± 2a (3) y = – x + a (4) y = – x + 2a
Q.27
The locus of the point of intersection of perpendicular tangent to the parabola x2 – 8x + 2y + 2 = 0 is (1) 2y – 15 = 0 (2) 2y + 15 = 0 (3) 2x + 9 = 0 (4) none of these
Q.28
The equation of the common tangent of the parabolas x2 = 108y and y2 =–12xare (1) 2x + 3y = 36 (2) 2x + 3y + 36 = 0 (3) 3x + 2y = 36 (4) 3x + 2y + 36 = 0
Q.29
If the line l x +my+n=0isatangenttotheparabolay2 = 4ax, then locus of its point of contact is (1)a straightline (2) a circle (3) a parabola (4)two straight lines
Q.30
Equation of the directrix of the parabola whose focus is (0, 0) and the tangent at the vertex is x – y + 1 = 0 is (1) x – y = 0 (2) x – y – 1 = 0 (3) x – y + 2 = 0 (4) x + y – 1 = 0 EXERCISE-2
Q.1 The eccentricity of the ellipse 9x2 + 5y2 – 30 y = 0 is(1) 1/3 (2) 2/3 (3) 3/4 Q.2
(4) None of these
Equation of the ellipse whose focusis (6,7)directrix is x + y + 2 = 0 and e = 1/ 3 is(1) 5x2 + 2xy + 5y2 – 76x – 88y + 506 = 0 (2) 5x2 – 2xy + 5y2 – 76x – 88y + 506 = 0 (3) 5x2 – 2xy + 5y2 + 76x + 88y – 506 = 0 (4) None of these
Q.3 The eccentricity of an ellipse
(1)
1
(2)
2
x2 a2
+
y2 b 2
= 1 whose latus rectum is half of its major axisis-
2
(3)
3
3
(4) None of these
2
Q.4
The equation of the ellipse whose centre is at origin and which passes through the points (– 3,1) and (2,–2) is(1) 5x2 + 3y2 = 32 (2)3x2 + 5y2 = 32 (3) 5x2 – 3y 2 = 32 (4) 3x2 + 5y2 + 32= 0
Q.5
The equation of the ellipse (referred toits axes as the axes of x and y respectively) which passes through the point (– 3, 1) and has eccentricity (1) 3x2 + 6y 2 = 33
2 5
, is-
(2) 5x2 + 3y 2 = 48
(3) 3x2 + 5y2 –32 = 0 (4) None of these
Q.6 Latus rectum of ellipse 4x2 + 9 y2 – 8x – 36 y + 4 = 0 is-
(1) 8/3
(2) 4/3
(3)
5
(4) 16/3
3
Q.7
If the distance between the foci of an ellipse be equal to its minor axis, then its eccentricity is(1) 1/2 (2) 1/ 2 (3) 1/3 (4) 1/ 3
Q.8
The equation of the ellipse whose centre is (2,– 3), one of the foci is (3,– 3) and the corresponding vertex is (4,– 3) is(1) (3)
( x 2) 2 3 x2 3
+
+
y2 4
( y 3) 2
=1
4
=1
(2)
( x 2) 2 4
+
( y 3) 2 3
(4) None of these
=1
Q.9 Eccentricity of the ellipse 4x2+y2 –8x+2y+1=0 is(1) 1/ 3 (2) 3/2 (3) 1/2 Q.10
(4) None of these
The eccentricityof anellipseis2/3, latusrectumis5 and centreis(0, 0). The equation of the ellipse is (1)
x2 81
y2 45
1
(2)
4x 2 81
4y 2
1
45
(3)
x2 9
y2 5
1
(4)
x2 81
y2 45
5
Q.11
The equation of the ellipse whose one of the vertices is (0, 7) and the corresponding directrix is y = 12, is(1) 95x2 + 144y2 = 4655 (2) 144x2 + 95y2 = 4655 (3) 95x2 + 144y2 = 13680 (4) None of these
Q.12
The equation of the ellipse whose foci are (± 5, 0)and one of its directrix is 5x = 36, is (1)
x
2
36
y
2
11
x2
1
(2) 6
y2 11
1
(3)
x
2
6
y
2
11
1
(4) None of these
Q.13 The position of the point (4,– 3) with respect to the ellipse 2x2 + 5y2 =20is(1) outside the ellipse (2) on the ellipse (3) on the major axis (4) None of these Q.14
x
y
x2
a
b
a2
If + = 2 touchesthe ellipse (1) 0
Q.15
(2) 90º
+
y2
=1, then its eccentric angle is equalto-
b 2
(3) 45º
(4) 60º
Find the equation of the tangent to the ellipse x2 + 2y2 = 4 atthe pointswhere ordinateis1. (1) x + 2 y –2 2 = 0 & x – 2 y +2 2 = 0 (2) x – 2 y –2 2 = 0 & x – 2 y +2 2 = 0 (3) x + 2 y +2 2 = 0 & x+ 2 y +2 2 = 0 (4)None of these
Q.16 Find the equations of tangents to the ellipse 9x2 + 16y2 = 144 which pass through the point (2,3). (1) y = 3 and y = –x + 5 (2) y = 5 and y = –x + 3 (3) y = 3 and y = x – 5 (4) None of these Q.17
The equation of the tangent at the point (1/4, 1/4) of the ellipse (1) 3x + y = 48
(2) 3x + y = 3
Q.19
x2 a2
y2 12
= 1, is-
(4) None of these +
y2 b 2
=1,if-
(2) p2 = a2 + b2 (4) None of these
The equation of the tangents to the ellipse 4x2 + 3y2 = 5 which are parallel to the line y = 3x + 7 are (1) y = 3x ±
Q.20
4
+
(3) 3x + y = 16
Q.18 The line x cos +ysin = p willbe a tangent to the conic
(1) p2=a2 sin2 + b2 cos2 (3) p2 = b 2 sin2 + a 2 cos2
x2
155 3
(2) y = 3x ±
155 12
Two perpendicular tangents drawn to the ellipse (1) x = a/e
(2) x2 + y 2 = 41
95
(3) y = 3x ± x2 25
+
y2 16
12
(4) None of these
= 1 intersect on the curve -
(3) x2 + y 2 = 9
(4) x2 – y 2 = 41
Q.21 The equation
x
2
10 a
+
(1) a < 4 Q.22
2
= 1 represents an ellipse if -
4a
(2) a > 4
(3) 4 < a < 10
2
x
2
k
+
y
2
h
2
=1
Let E be the ellipse
(2) x2 9
x
2
+
2
k
y2
+
y
2
k h
2
= 1 (3)
π
Q.26
(2)
8
Q.29
2
2
h k
=1
(4)
x
2 2
k
+
y
2
k h
2
=1
4
(2) Q lies outside both C and E (4) P lies inside C but outside E
(3)
3
3
(4)
4
2 3
(4) None
If the minor axis of an ellipse subtends an angle 60° at each focus then the eccentricity of the ellipse is (2) 1 / 2
(3) 2 / 3
(4) None
LL isthe latus rectum of an ellipse and SLL is an equilateral triangle. The eccentricityofthe ellipse is(1) 1 / 5
Q.28
+
2
k
y
The common tangent of x2 + y2 =4and2x2 + y2 =2is(1) x + y + 4 = 0 (2) x – y + 7 = 0 (3) 2x + 3y + 8 = 0
(1) 3 / 2 Q.27
2
Eccentric angle of a point on the ellipse x2 + 3y2 = 6 at a distance 2 units from the centre of the ellipse is (1)
Q.25
x
= 1 and C be the circle x2 + y2 = 9. Let P and Q be the points
(1, 2) and (2, 1) respectively. Then (1) Q lies inside C but o utside E (3) P lies inside both C and E Q.24
(4) a > 10
If the focal distance of an end of the minor axis of an ellipse (referred to its axes as the axes of x and y respectively) is k and the distance between its foci is 2h, then its equation is(1)
Q.23
y
(2) 1 / 3
(3) 1 / 2
(4) 2 / 3
If P is a point on the ellipse of eccentricity e and A,A are the vertices and S, S are the focii then SPS : APA = (1) e3 (2) e2 (3) e (4) 1/e The length of the common chord of the ellipse the circle (x –1)2 + (y –2)2 = 1 i s (1) 0 (2) 1
Q.30 If any tangent to the ellipse
(1)
a 2 b 2
x2 a2
( x 1) 2 9
( y 2) 2 4
1 and
(3) 3 y2
b 2
(4) 8
1 intercepts equal lengths on the axes, then =
(2) a2 + b2
(3) (a2 + b2)2
(4) None of these
EXERCISE-3
Q.1
Ifthe latus rectumofan hyperbola be8 andeccentricity be (1) 4x2 – 5y 2 = 100 (3) 4x2 + 5y 2 = 100
Q.2 Foci of the hyperbola
(1) (5, 2); (–5, 2)
3 5
, then the equation ofthe hyperbola is-
(2) 5x2 – 4y 2 = 100 (4) 5x2 + 4y 2 = 100 x2 16
–
( y 2) 2 9
=1are
(2) (5, 2); (5, –2)
(3) (5, 2); (–5, –2)
(4)None of these
Q.3
Equation of the hyperbola with eccentricity 3/2 and foci at (± 2, 0) is(1)
x2 4
–
y2
4
=
5
(2)
9
x2 9
–
y2
4
=
9
(3)
9
x2
–
4
y2
=1
9
(4) None of these
Q.4
If the centre, vertex and focus of a hyperbola be (0, 0), (4, 0) and (6, 0) respectively, then the equation of the hyperbola is(1) 4x2 – 5y 2 = 8 (2) 4x2 – 5y 2 = 80 (3) 5x2 – 4y 2 = 80 (4) 5x2 – 4y 2 = 8
Q.5
The eccentricityof the hyperbola whoselatus rectum is 8 and conjugate axis isequal to half the distance between the foci is(1)
Q.6
4
(2)
3
4
2
(3)
3
(4) None of these
3
The equation of the hyperbola whose foci are (6, 5) , (– 4, 5) and eccentricity 5/4 is(1) (3)
( x 1) 2 16 ( x 1) 2 9
– –
( y 5) 2 9 ( y 5) 2
Q.7 The equation
16 x
(2)
=1
(4) None of these
+
y
16
–
y2
=1
2
12
x2
9
=1
2
8
= 1 represents
(1) a hyperbola if < 8 (3) a hyperbola if 8 < < 12
(2) an ellipse if > 8 (4) None of these
Q.8 Ifeande be the eccentricities of two conics S and S such that e2 + e2 =3,thenbothSandS are(1) ellipse (2) parabolas (3) hyperbolas (4) None of these Q.9
The equation of the conic with focus at (1, –1), directrix along x – y + 1 = 0 and with eccentricity 2 is(1) x2 – y 2 = 1 (2) xy = 1 (3) 2xy – 4x + 4y + 1 = 0 (4) 2xy + 4x – 4y – 1 = 0
Q.10 The latus rectum of a hyperbola
(1) 4/5 Q.11
Q.12
x
2
–
16
y2
1
=1is4 . Its eccentricity e =
p
2
(2) 5/4
(3) 3/4
(4) 4/3
Consider the set of hyperbola xy = k, k R. Let e1 be the eccentricity when k = 4 and e2 be the eccentricity when k = 9. Then e12 + e22 = (1) 2 (2) 3 (3) 4 (4) 1 x2
The eccentricity of the hyperbola –
a2
(1) e = +
a 2 b 2 a2
(2) e = +
+
y2 b 2
a 2 b 2 a2
=1isgivenby(3) e = +
Q.13 The equation of a tangent parallel to y = x drawn to
(1) x – y + 1 = 0
(2) x – y + 2 = 0
x2 3
–
b 2 a 2 a2
y2 2
(4) e = +
a 2 b 2 b 2
=1is-
(3) x + y – 1 = 0
(4) x – y + 2 = 0
Q.14
The equation of tangents to the hyperbola x2 – 4y2 = 36 which are perpendicular to the line x–y+4=0 (1) y = – x + 3 3 (2) y = x – 3 3 (3) y = – x ± 2 (4) N one of these
Q.15
The line y = x + 2 touches the hyperbola 5x2 – 9y2 = 45 at the point(1) (0, 2) (2) (3, 1) (3) (–9/2, –5/2)
(4) None of these
Q.16
The value of m for which y = mx+ 6 is a tangent to thehyperbola (1)
Q.17
17
20
(2)
20
(3)
17
x
100
3 20
x
x2 4
y2
1 (2)
12
x2 4
y2 12
1 (3)
x2 12
49
=1is20 3
9
y2 4
1 (4)
x2 12
–
y2 4
= 1. If equation of QR is
(4) None of these
y2 4
x2 25
y2 9
1 andtheeccentricityis2,is-
1
A common tangent to 9x2 – 16y2 = 144 and x2 + y2 = 9 is (1) y = (3) y = 2
3
15
x+
7
3
7
x+
7
(2) y = 3
(1) ± 3
2 7
x+
15 7
(4) none of these
15 7
Q.21 If the focii of the ellipse
x2 k 2 a 2
+
y2 a2
= 1 and the hyperbola
(2) ± 2
(3)
x2 a2
–
y2 a2
3
= 1 coincides then value of k = (4)
2
If the latus rectum subtends a right angle at the centre of the hyperbola then its eccentricity is (1) e = ( 13 )/ 2
Q.23 The equation x=
(2) e = ( 5 –1) /2 e t e t 2
(1) an ellipse
Q.24 The ellipse
x2 25
;y=
e t e t 2
(3) e = ( 5 + 1)/2
(2) a parabola
+
y2 16
(4) e = ( 3 + 1)/2
; t R represents
= 1 and the hyperbola
(3) a hyperbola x2 25
(1) centre only (3)centre, foci and vertices
Q.25
2
(3) (3, 4)
Theequationofthehyperbolawhosefociarethefocioftheellipse (1)
Q.22
2
Equation of one of common tangent to parabola y2 = 8x and hyperbola 3x2 – y2 =3is(1) 2x –y –1 =0 (2) 2x –y +1 =0 (3) y +2x +1 =0 (4) y – 2x + 1= 0
4x – 3y – 6 = 0, then coordinates of P is(1) (2, 6) (2) (6, 2)
Q.20
y
–
(4)
Q.18 If PQ and PR are tangents drawn from a point P to the hyperbola
Q.19
2
If e1, e2 are the eccentricities of the ellipse then the relation between e1 and e2 is (1) 3e12 + e22 = 2 (2) e12 + 2e22 = 3
–
y2 16
(4) a circle
= 1 have in common -
(2) centre, foci and directrices (4)centre and vertices only x2 18
+
y2 4
= 1 and the hyperbola
(3) 2e12 + e22 = 3
x2 9
–
y2 4
= 1 respectively,,
(4) e12 + 3e22 = 2
EXERCISE-4
Assertion & Reason type questions : All questions are Assertion & Reason type questions. Each of these questions contains two statements : Statement-1 (Assertion) and Statement-2 (Reason).Answer these questions from the following four option.
Q.1
(A)
If both Statement- 1 and Statement- 2 are true, and Statement - 2 is the correct explanation of Statement– 1.
(B)
If both Statement - 1 and Statement - 2 are true but Statement - 2 is not the correct explanation of Statement – 1.
(C)
If Statement - 1 is true but Statement - 2 is false.
(D)
If Statement - 1 is false but Statement- 2 is true.
Statement-1 : The curve 9y2 – 16x – 12y – 57 = 0 is symmetric about line 3y = 2. Statement-2 : A parabola is symmetric about it's axis.
(1) A Q.2
(3) C
(4) D
Statement-1 : Two perpendicular tangents of parabola y2 = 16x always meet on x + 4 = 0. Statement-2 : Two perpendicular tangents of a parabola, always meets on axis.
(1) A Q.3
(2) B
(2) B
(3) C
(4) D
Statement-1 : If4 and 3 are lengthof two focal segmentsoffocal chord of parabola y2 = 4ax than latus 48 rectum of parabol will be units. 7 2l 1l 2 Statement-2 : If l 1 and l 2 are length of focal segments of focal chord than its latus rectum is . l 1 l 2
(1) A Q.4
(2) B
(3) C
(4) D
Statement-1 : Let (x1, y1)and(x2, y2) are the ends of a focal chord of y2 = 4x then4x1x2 + y1y2 = 0. Statement-2 : PSQ is the focal of a parabola with focus S and latus rectum thenSP+SQ=2l .
(1) A Q.5
(2) B
(3) C
(4) D
Statement-1 : PQ is the double ordinate of a parabola. QR is the focal chord. Then PR passes through
the foot of the directrix. Statement-2 : The chord joiningthe points t1 andt2 arey2 = 4axpasses through the foot of its directrix then t1t2 = 1. (1) A (2) B (3) C (4) D Q.6
Statement-1 : PQ is a focal chord of a parabola. Then the tangent at P to the parabola isparallel to the
normal at Q. Statement-2 : If P(t1) and Q(t2) are the ends of a focal chord of the parabola y2 = 4ax, then t1t2 = – 1 . (1) A (2) B (3) C (4) D
Q.7
Statement-1 : The tangents drawn to the parabola y2 = 4ax at the ends of any focal chord intersect on the directrix. Statement-2 : The point of intersection of the tangents at drawn at P(t1) and Q(t2) are the parabola y2 = 4ax is {at1t2, a (t1 + t 2)} (1) A (2) B (3) C (4) D
Q.8
Statement-1 : The circum circle of the triangle formed by any three tangents to a parabola, passes through thefocus. Statement-2 : The feet of the altitudes from the focus of a parabola to the sides of the triangle formed by any three tangents to the parabola, are collinear. (1) A (2) B (3) C (4) D
Q.9
Statement-1: From a point (5, ) perpendicular tangents are drawn to the ellipse
then = ±4.
x
25
Statement- 2 : The locus of the point of intersectionof perpendicular tangent to theellipse
(1) A
is x 2 + y2 = 41. (2) B
(3) C
2
x
+
y
16
2
25
2
+
y
=1
2
16
=1
(4) D
Q.10
Statement-1: The semi minoraxis isthe geometric mean between SA and SA where S, S are the foci of an ellipse and A is its vertex. Statement-2 : The product of perpendiculars drawn from the foci to any tangent of ellipse is equal to squareas the semi minoraxis. (1) A (2) B (3) C (4) D
Q.11
Statement-1: PS P is a focal chord of
x2 25
y2 16
1 . If SP = 8 then SP = 2.
Statement-2: The semi latus-rectum of an ellipse is the harmonic mean between the segments of a focal chord. (1) A (2) B (3) C (4) D Q.12
Statement-1: If P(x1, y1) is a point on b2x2 + a2y2 = a2 b2 then area SPS = ae Statement-2: A tangent to
x2
y2
a
b 2
– 2
= 1 meets the major axisin P and Q then
a 2 x 12
a2 CP
– 2
b 2 CQ 2
= 1, where
C is the centre of the conic. Which of the statements is correct? (1) both 1 and 2 (2) only 1 (3) only 2 (4) neither 1 nor 2 Q.13
Statement-1: The length of latus-rectum of the hyperbola x2 – y2 = a2 is 2a. Statement-2: The length of the latus-rectum of the hyperbola xy = 2c2 is 4c. Whichof the statements is correct? (1) both 1 and 2 (2) only 1 (3) only 2 (4) neither 1 nor 2
Q.14
Statement-1: The conic 16x2 –3y2 –32x + 12y – 44 = 0 represent a hyperbola. Statement-2: The square of the coefficient of xy is greater than the product of the coefficient of x2 & y 2 and 0. (1) A (2) B (3) C (4) D
Q.15
Statement-1: The latus-rectumof the hyperbola x2 – y2 = a2 is equal to the length of its major axis. 2 2
2 2
2
2
Statement-2: The semi latusrectum of the ellipse b x + a y = a b is equal to
(1) A
(2) B
(3) C
(4) D
b
2
a
.
Q.16
Statement-1: If a point(x1, y1) lies in the region II of the hyperbola
then
y12 – < 0. a 2 b 2
Q.17
a2
y2 b 2
1 , shown in the figure,
x 12
Statement-2: If a point P(x1, y1) lies outside the hyperbola
(1) A
x2
(2) B
(3) C
x2 a2
y2
x 12
b
a2
1 then 2 (4) D
–
y12 b 2
< 1.
Statement-1: The equation x2 + 2y2 + xy + 2x + 3y + 1 = 0 can never represent a hyperbola. Statement-2: The general equation of second degree represents a hyperbola if h2 >ab. (1) A (2) B (3) C (4) D Paragraph for Question no. 18 to 20
Variable tangent drawnto ellipse
x2 a2
y2
+
b 2
= 1 (a > b) intersects major and minor axis at points A & B in
first quadrant then Q.18 Area of OAB is minimum when =
(1) Q.19
(2)
3
(3)
6
Minimumvalueof lengthABis (1) 2b (2) a + b
Q.20 Locus of centroid of OAB is
(1) 1
a2 x2
(4)
4
(3) ab +
b 2 y
2
2
(4) b
= k 2 then k =
(2) 2
(3) 3
(4) 4
Paragraph for Question no. 21 to 22
AparabolaP:y 2 = 8x, ellipse E : Q.21
x2 4
+
y2 15
= 1.
Equation of a tangent common to both the parabola P and the ellipse E is (1) x – 2y + 8 = 0 (2) 2x – y + 8 = 0 (3) x + 2y – 8 = 0 (4) 2x – y – 8 = 0
Q.22 Point of contact of a common tangent to P and E on the ellipse is 1 15
(1) 2 , 4
1 15
1
(2) 2 , 4
15
1
(3) 2 , 2
15
(4) 2 , 2
Paragraph for Question no. 23 to 24
If parametric equation of hyperbola is x = Q.23
2
&y=
e t et 3
then-
Eccentricity of hyperbola is 13
(1) Q.24
e t e t
2
13
(2)
3
2
(3)
3 2
(4) 13
Eccentric angle of point 2, on hyperbola 3 (1)
6
(2)
4
(3)
3
(4) None of these
Paragraph for Question no. 25 to 27
A equilateral triangle is inscribed in the parabola y2 = 8x. If one vertex of the triangle is at the vertex of parabola then Q.25 Length of side of the triangle is
(1) 8 3
(2) 16 3
(3) 4 3
(4) 8
(2) 48 3
(3) 192 3
(4) none of these
(3) 16
(4) 32
Q.26 Area of the triangle is
(1) 64 3 Q.27
Radiusof the circum-circle of the triangle is (1) 4 (2) 8
Paragraph for Question no. 28 to 30 Conic possesses enormous properties which can be probed by taking standard forms. Unlike circle these properties rarelyfollowby geometricalconsiderations. Most of theproperties of conicsare proved analytically. For example, the properties of a parabola can be proved by taking its standard equation y2 = 4ax and a point (at2, 2at) on it. Q.28 If the tangent and normal at any point P on the parabola whose focus is S, meets its axis in T andG respectively, then (1) PG = PT (2) S is mid-point of T and G (3) ST = 2SG (4) none of these Q.29
The angle between the tangents drawn at the extremities of a focal chord must be (1) 30° (2) 60° (3) 90° (4) 120°
Q.30
If the tangent at any point P meets the directrix at K, then KSP must be (1) 30° (2) 60° (3) 90° (4) none of these Match the column
Q.31 The parabola y2 = 4ax has a chord AB joing points A at12 , 2at1 and B at 22 , 2at 2 . Column-I
(1)
AB is a normal chord if
Column-II
(P)
t = – t – 2
(2)
AB is a focal chord
(Q)
t = 2
(3)
AB subtends 90° at point (0, 0) if
(R)
t = 2
(S)
1
2
4 t1
1 t1
t = – t – 2
1
1
2 t1
Q.32
(1) (2) (3) (4)
Column-I The focus of the parabola (y – 4)2 = 12 (x – 2) The vertex of the parabola y2 – 5x – 4y + 9 = 0 The foot of the directrix of the parabola x2 + 8y = 0 The vertex of the parabola x2 + 6x – 2y + 11 = 0
Column-II (P) (1, 2) (Q) (–3, 1) (R) (3, 0) (S) (5, 4) (T) (2, 0)
(1) (2) (3) (4)
Column-I The directrix of the parabola y2 – 2y + 8x – 23 = 0 The equation to the latus rectum of x2 – 2x – 4y – 3 = 0 The axis of the parabola x2 + 8x + 12y + 4 = 0 is Equation to the tangent at vertex of y2 – 6y – 12x – 15 = 0
Column-II (P) x+ 2= 0 (Q) y– 5= 0 (R) x+4=0 (S) x– 5= 0 (T) y–2=0
Q.33
Q.34
Column-I
Column-II
25
(1)
The length of the latus rectum of the parabola whose
(P)
(2) (3) (4)
focus is (4, 5) and the vertex (3, 6) The line 2x – y + 2 = 0 touches the parabola y2 = 4px. Then p = The length of the focal chord of y2 = 8x drawn through (8, 8)is y = 3x + c is a tangent to y2 = 12x then c =
(Q) (R) (S)
4 –2 1
(T)
4 2
2
EXERCISE-1 Q.1 Q.6 Q.11 Q.16 Q.21 Q.26
1 3 3 3 2 2
Q.2 Q.7 Q.12 Q.17 Q.22 Q.27
3 1 3 3 1 1
Q.3 Q.8 Q.13 Q.18 Q.23 Q.28
2 4 2 2 3 2
Q.4 Q.9 Q.14 Q.19 Q.24 Q.29
3 3 4 4 4 3
Q.5 Q.10 Q.15 Q.20 Q.25 Q.30
2 1 3 2 2 3
EXERCISE-2 Q.1 Q.6 Q.11 Q.16
2 1 2 1
Q.2 Q.7 Q.12 Q.17
2 2 1 4
Q.3 Q.8 Q.13 Q.18
1 2 1 3
Q.4 Q.9 Q.14 Q.19
2 3 3 2
Q.5 Q.10 Q.15 Q.20
3 2 1 2
Q.21 Q.26
1 1
Q.22 Q.27
2 2
Q.23 Q.28
4 3
Q.24 Q.29
3 1
Q.25 Q.30
4 1
Q.4 Q.9 Q.14 Q.19 Q.24
3 3 1 2 4
Q.5 Q.10 Q.15 Q.20
3 2 3 2
Q.5 Q.10 Q.15 Q.20 Q.25 Q.30
1 1 1 3 2 3
EXERCISE-3 Q.1 Q.6 Q.11 Q.16 Q.21 Q.25
2 1 3 1 1 3
Q.2 Q.7 Q.12 Q.17 Q.22
1 3 4 23 3
Q.3 Q.8 Q.13 Q.18 Q.23
1 3 1 2 3
EXERCISE-4 Q.1 1 Q.2 2 Q.6 1 Q.7 2 Q.11 1 Q.12 3 Q.16 4 Q.17 4 Q.21 1 Q.22 2 Q.26 3 Q.27 3 Q.31 (1) S ; (2) R ; (3) Q Q.33 (1) S ; (2) T ; (3) R ; (4) P
Q.3 Q.8 Q.13 Q.18 Q.23 Q.28 Q.32 Q.34
3 Q.4 3 1 Q.9 1 1 Q.14 1 3 Q.19 2 4 Q.24 3 2 Q.29 3 (1) S ; (2) P ; (3) Q ; (4) T (1) T ; (2) Q ; (3) P ; (4) S
