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CONIC SECTION(PARABOLA, ELLIPSE & HYPERBOLA)
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 vertex and 3x – 4y + 2 = 0 be the directrix of a parabol, then the length of its latus rectumis
(1)5
4(2)
5
2(3)
5
8(4)
5
1
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 is 2x – 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 = 12y to the ends of itsrectum 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)
9
2,
3
1(2)
2
1,
3
1(3)
2
1,
3
1(4)
2
1,
3
1
Q.8 The focus of the parabola y2 – x – 2y + 2 = 0 is
(1) (1, 2) (2)
0,
4
1(3)
1,
4
3(4)
1,
4
5
Q.9 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 (4) y – 2 = 0
Q.10 The point of intersection of the latus rectum and axes of the parabola y2 + 4x + 2y – 8 = 0 is
(1)
1,
4
5(2)
2
5,
5
7(3)
1,
4
9(4) none of these
Q.11 Which of the following is not the equation of parabola(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, thenits 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° anglewith x-axis is
(1)2
3(2)
2
3(3) 8 3 (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. Then the 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
(1) c = am –
m
a(2) c =
m
a(3) c =
m
a(4) c = am +
m
a
Q.20 For what value of k, the line 2y – x + k = 0 touches the parabola x2 + 4y = 0
(1) 2 (2)2
1(3) – 2 (4)
2
1
Q.21 The point where the line x + y = 1 touches the parabola y = x – x2, is
(1)
2
1,
2
1(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 on the curve y2 = x the tangent at which makes an angle of 45° with x-axis will be given by
(1)
2
1,
2
1(2)
4
1,
2
1(3) (2, 4) (4)
2
1,
4
1
Q.24 The equation of the common tangents to the parabolas y2 = 4x and x2 = 32y is(1) x + 2y = 4 (2) x = 2y + 4 (3) x = 2y – 4 (4) x + 2y + 4 = 0
Q.25 If m1 and m2 are slopes of the two tangents that are drawn from (2, 3) to the parabola y2 = 4x then value
of
21 m
1
m
1=
(1) – 3 (2) 3 (3)3
2(4)
2
3
Q.26 The equation of common tangent to the circle x2 + y2 = 2a2 and parabola y2 = 8ax is(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 = – 12x are(1) 2x + 3y = 36 (2) 2x + 3y + 36 = 0 (3) 3x + 2y = 36 (4) 3x + 2y + 36 = 0
Q.29 If the line lx + my + n = 0 is a tangent to the parabola y2 = 4ax, then locus of its point of contact is(1) a straight line (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 isx – y + 1 = 0 is(1) x – y = 0 (2) x – y – 1 = 0 (3) x – y + 2 = 0 (4) x + y – 1 = 0
EXERCISE-2Q.1 The eccentricity of the ellipse 9x2 + 5y2 – 30 y = 0 is-
(1) 1/3 (2) 2/3 (3) 3/4 (4) None of these
Q.2 Equation of the ellipse whose focus is (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 2
2
a
x+ 2
2
b
y= 1 whose latus rectum is half of its major axis is-
(1)2
1(2)
3
2(3)
2
3(4) None of these
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 – 3y2 = 32 (4) 3x2 + 5y2 + 32= 0
Q.5 The equation of the ellipse (referred to its axes as the axes of x and y respectively) which passes through
the point (– 3, 1) and has eccentricity5
2 , is-
(1) 3x2 + 6y2 = 33 (2) 5x2 + 3y2 = 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)3
5(4) 16/3
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 correspondingvertex is (4,– 3) is-
(1)3
)2x( 2+
4
)3y( 2= 1 (2)
4
)2x( 2+
3
)3y( 2= 1
(3)3
x 2
+4
y2
= 1 (4) None of these
Q.9 Eccentricity of the ellipse 4x2+y2–8x+2y+1=0 is-(1) 1/ 3 (2) 3/2 (3) 1/2 (4) None of these
Q.10 Theeccentricityof an ellipse is 2/3, latus rectum is5 and centre is (0, 0). The equation of the ellipse is -
(1) 145
y
81
x 22
(2) 145
y4
81
x4 22
(3) 15
y
9
x 22
(4) 545
y
81
x 22
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) 111
y
36
x 22
(2) 111
y
6
x 22
(3) 111
y
6
x 22
(4) None of these
Q.13 The position of the point (4,– 3) with respect to the ellipse 2x2 + 5y2 = 20 is-(1) outside the ellipse (2) on the ellipse(3) on the major axis (4) None of these
Q.14 Ifa
x+
b
y= 2 touchestheellipse 2
2
a
x+ 2
2
b
y=1, then its eccentric angle is equal to-
(1) 0 (2) 90º (3) 45º (4) 60º
Q.15 Find the equation of the tangent to the ellipse x2 + 2y2 = 4 at the points where ordinate is 1.(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 ellipse4
x 2
+12
y2
= 1, is-
(1) 3x + y = 48 (2) 3x + y = 3 (3) 3x + y = 16 (4) None of these
Q.18 The line x cos+ y sin= p will be a tangent to the conic 2
2
a
x+ 2
2
b
y= 1, if-
(1) p2=a2 sin2 + b2 cos2 (2) p2 = a2 + b2
(3) p2 = b2 sin2 + a2 cos2 (4) None of these
Q.19 The equation of the tangents to the ellipse 4x2 + 3y2 = 5 which are parallel to the liney = 3x + 7 are
(1) y = 3x ±3
155(2) y = 3x ±
12
155(3) y = 3x ±
12
95(4) None of these
Q.20 Two perpendicular tangents drawn to the ellipse25
x 2
+16
y 2
= 1 intersect on the curve -
(1) x = a/e (2) x2 + y2 = 41 (3) x2 + y2 = 9 (4) x2 – y2 = 41
Q.21 The equationa10
x 2
+
a4
y2
= 1 represents an ellipse if -
(1) a < 4 (2) a > 4 (3) 4 < a < 10 (4) a > 10
Q.22 If the focal distance of an end of the minor axis of an ellipse (referred to its axes as the axes of x and yrespectively) is k and the distance between its foci is 2h, then its equation is-
(1) 2
2
k
x+ 2
2
h
y= 1 (2) 2
2
k
x+ 2
2
hk
y
= 1 (3) 2
2
k
x+ 2
2
kh
y
= 1 (4) 2
2
k
x+ 2
2
hk
y
= 1
Q.23 Let E be the ellipse9
x 2
+4
y2
= 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 outside E (2) Q lies outside both C and E(3) P lies inside both C and E (4) P lies inside C but outside E
Q.24 Eccentric angle of a point on the ellipse x2 + 3y2 = 6 at a distance 2 units from the centreof the ellipse is -
(1)8
π(2)
3
(3)
4
3(4)
3
2
Q.25 The common tangent of x2 + y2 = 4 and 2x2 + y2 = 2 is-(1) x + y + 4 = 0 (2) x – y + 7 = 0 (3) 2x + 3y + 8 = 0 (4) None
Q.26 If the minor axis of an ellipse subtends an angle 60° at each focus then the eccentricity of the ellipse is -
(1) 2/3 (2) 2/1 (3) 3/2 (4) None
Q.27 LL is the latus rectum of an ellipse andSLL is an equilateral triangle.The eccentricityof the ellipse is -
(1) 5/1 (2) 3/1 (3) 2/1 (4) 3/2
Q.28 If P is a point on the ellipse of eccentricity e andA,A are the vertices and S, S are the focii thenSPS: APA =(1) e3 (2) e2 (3) e (4) 1/e
Q.29 The length of the common chord of the ellipse 14
)2y(
9
)1x( 22
and
the circle (x –1)2 + (y –2)2 = 1 is(1) 0 (2) 1 (3) 3 (4) 8
Q.30 If any tangent to the ellipse 1b
y
a
x2
2
2
2
intercepts equal lengths on the axes, then =
(1) 22 ba (2) a2 + b2 (3) (a2 + b2)2 (4) None of these
EXERCISE-3
Q.1 If the latus rectum of an hyperbola be8 andeccentricitybe5
3, then theequation of the hyperbola is-
(1) 4x2 – 5y2 = 100 (2) 5x2 – 4y2 = 100(3) 4x2 + 5y2 = 100 (4) 5x2 + 4y2 = 100
Q.2 Foci of the hyperbola16
x2
–9
)2y( 2= 1 are
(1) (5, 2); (–5, 2) (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)4
x 2
–5
y2
=9
4(2)
9
x 2
–9
y2
=9
4(3)
4
x 2
–9
y2
= 1 (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 equationof the hyperbola is-(1) 4x2 – 5y2 = 8 (2) 4x2 – 5y2 = 80 (3) 5x2 – 4y2 = 80 (4) 5x2 – 4y2 = 8
Q.5 The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distancebetween the foci is-
(1)3
4(2)
3
4(3)
3
2(4) None of these
Q.6 The equation of the hyperbola whose foci are (6, 5) , (– 4, 5) and eccentricity 5/4 is-
(1)16
)1x( 2–
9
)5y( 2= 1 (2)
16
x 2
–9
y2
= 1
(3)9
)1x( 2–
16
)5y( 2= 1 (4) None of these
Q.7 The equation12
x 2
+8
y2
= 1 represents
(1) a hyperbola if < 8 (2) an ellipse if > 8(3) a hyperbola if 8 < < 12 (4) None of these
Q.8 If e and e be the eccentricities of two conics S and S such that e2 + e2 = 3, then both S and S 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 witheccentricity 2 is-(1) x2 – y2 = 1 (2) xy = 1(3) 2xy – 4x + 4y + 1 = 0 (4) 2xy + 4x – 4y – 1 = 0
Q.10 The latus rectum of a hyperbola16
x 2
–p
y 2
=1 is 42
1. Its eccentricity e =
(1) 4/5 (2) 5/4 (3) 3/4 (4) 4/3
Q.11 Consider the set of hyperbola xy = k, k R. Let e1 be the eccentricity when k = 4 and e2 be theeccentricity when k = 9. Then e1
2 + e22 =
(1) 2 (2) 3 (3) 4 (4) 1
Q.12 The eccentricity of the hyperbola – 2
2
a
x+ 2
2
b
y= 1 is given by -
(1) e = + 2
22
a
ba (2) e = + 2
22
a
ba (3) e = + 2
22
a
ab (4) e = + 2
22
b
ba
Q.13 The equation of a tangent parallel to y = x drawn to3
x2
–2
y2
= 1 is-
(1) x – y + 1 = 0 (2) x – y + 2 = 0 (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 linex – y + 4 = 0(1) y = – x + 3 3 (2) y = x – 3 3 (3) y = – x ± 2 (4) None 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 the hyperbola100
x 2
–49
y2
= 1 is-
(1)20
17(2)
17
20(3)
20
3(4)
3
20
Q.17 Equation of one of common tangent to parabola y2 = 8x and hyperbola 3x2 – y2 = 3 is-(1) 2x–y–1=0 (2) 2x–y+1=0 (3) y+2x+1=0 (4) y – 2x+ 1 = 0
Q.18 If PQ and PR are tangents drawn from a point P to the hyperbola9
x 2
–4
y2
= 1. If equation of QR is
4x – 3y – 6 = 0, then coordinates of P is-(1) (2, 6) (2) (6, 2) (3) (3, 4) (4) None of these
Q.19 Theequationofthehyperbolawhosefociarethefocioftheellipse 19
y
25
x 22
andtheeccentricity is2,is-
(1) 112
y
4
x 22
(2) 112
y
4
x 22
(3) 14
y
12
x 22
(4) 14
y
12
x 22
Q.20 A common tangent to 9x2 – 16y2 = 144 and x2 + y2 = 9 is -
(1) y =7
3x +
7
15(2) y = 3
7
2x +
7
15
(3) y = 27
3x + 715 (4) none of these
Q.21 If the focii of the ellipse 22
2
ak
x+ 2
2
a
y= 1 and the hyperbola 2
2
a
x– 2
2
a
y= 1 coincides then value of k =
(1) ± 3 (2) ± 2 (3) 3 (4) 2
Q.22 If the latus rectum subtends a right angle at the centre of the hyperbola then its eccentricity is
(1) e = ( 13 )/ 2 (2) e = ( 5 –1) /2 (3) e = ( 5 + 1)/2 (4) e = ( 3 + 1)/2
Q.23 The equation x=2
ee tt ;y=
2
ee tt ; tR represents
(1) an ellipse (2) a parabola (3) a hyperbola (4) a circle
Q.24 The ellipse25
x 2
+16
y2
= 1 and the hyperbola25
x 2
–16
y2
= 1 have in common -
(1) centre only (2) centre, foci and directrices(3) centre, foci and vertices (4) centre and vertices only
Q.25 If e1, e2 are the eccentricities of the ellipse18
x 2
+4
y2
= 1 and the hyperbola9
x 2
–4
y2
= 1 respectively,,
then the relation between e1 and e2 is(1) 3e1
2 + e22 = 2 (2) e1
2 + 2e22 = 3 (3) 2e1
2 + e22 = 3 (4) e1
2 + 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.
(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.
Q.1 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 (2) B (3) C (4) D
Q.2 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 (2) B (3) C (4) D
Q.3 Statement-1 : If 4 and 3 are length of two focal segments of focal chord of parabola y2 = 4ax than latus
rectum of parabol will be7
48units.
Statement-2 : If l1and l
2are length of focal segments of focal chord than its latus rectum is
21
212
ll
ll
.
(1)A (2) B (3) C (4) D
Q.4 Statement-1 : Let (x1, y
1) and (x
2, y
2) are the ends of a focal chord of y2 = 4x then 4x
1x
2+ y
1y
2= 0.
Statement-2 : PSQ is the focal of a parabola with focus S and latus rectum then SP + SQ = 2l.
(1)A (2) B (3) C (4) D
Q.5 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 joining the points t1and t
2are y2 = 4ax passes 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 is parallel to the
normal at Q.
Statement-2 : If P(t1) and Q(t
2) are the ends of a focal chord of the parabola y2 = 4ax, then t
1t2= – 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 onthe directrix.
Statement-2 : The point of intersection of the tangents at drawn at P(t1) and Q(t
2) are the parabola
y2 = 4ax is {at1t2, a (t
1+ 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, passesthrough the focus.
Statement-2 : The feet of the altitudes from the focus of a parabola to the sides of the triangle formedby 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 ellipse25
x 2
+16
y2
= 1
then = ±4.
Statement- 2 : The locus of the point of intersection of perpendicular tangent to the ellipse25
x 2
+16
y2
= 1
is x2 + y2 = 41.(1)A (2) B (3) C (4) D
Q.10 Statement-1: The semi minor axis is the geometric mean between SA and SA where S, S are the fociof an ellipse andAis its vertex.
Statement-2 : The product of perpendiculars drawn from the foci to any tangent of ellipse is equal tosquare as the semi minor axis.
(1)A (2) B (3) C (4) D
Q.11 Statement-1: PS P is a focal chord of 116
y
25
x 22
. If SP = 8 then SP = 2.
Statement-2: The semi latus-rectum of an ellipse is the harmonic mean between the segments of afocal chord.
(1)A (2) B (3) C (4) D
Q.12 Statement-1: If P(x1, y
1) is a point on b2x2 + a2y2 = a2b2 then area SPS = ae 2
12 xa
Statement-2: A tangent to2
2
a
x–
2
2
b
y= 1 meets the major axis in P and Q then
2
2
CP
a–
2
2
CQ
b= 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.Which of 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 & y2 and 0.(1)A (2) B (3) C (4) D
Q.15 Statement-1: The latus-rectum of the hyperbola x2– y2 = a2 is equal to the length of its major axis.
Statement-2: The semi latusrectum of the ellipse b2x2 + a2y2 = a2 b2 is equal toa
b 2
.
(1)A (2) B (3) C (4) D
Q.16 Statement-1: If a point (x1, y
1) lies in the region II of the hyperbola 1
b
y
a
x2
2
2
2
, shown in the figure,
then 2
21
2
21
b
y–
a
x< 0.
Statement-2: If a point P(x1, y
1) lies outside the hyperbola 1
b
y
a
x2
2
2
2
then 2
21
a
x– 2
21
b
y< 1.
(1)A (2) B (3) C (4) D
Q.17 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 tangentdrawntoellipse 2
2
a
x+ 2
2
b
y= 1 (a > b) intersects major and minor axis at pointsA& B in
first quadrant then
Q.18 Area ofOAB is minimum when=
(1)3
(2)
6
(3)
4
(4)
2
Q.19 Minimumvalueof lengthABis
(1) 2b (2) a + b (3) ab (4) b
Q.20 Locus of centroid of OAB is 2
2
x
a+ 2
2
y
b= k2 then k =
(1) 1 (2) 2 (3) 3 (4) 4
Paragraph for Question no. 21 to 22
A parabola P : y2 = 8x, ellipse E :4
x 2
+15
y2
= 1.
Q.21 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)
4
15,
2
1(2)
4
15,
2
1(3)
2
15,
2
1(4)
2
15,
2
1
Paragraph for Question no. 23 to 24
If parametric equation of hyperbola is x =2
ee tt & y =
3
ee tt then-
Q.23 Eccentricity of hyperbola is
(1)2
13(2)
3
13(3)
2
3(4) 13
Q.24 Eccentric angle of point
3
2,2 on hyperbola
(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 ofparabola then
Q.25 Length of side of the triangle is
(1) 8 3 (2) 16 3 (3) 4 3 (4) 8
Q.26 Area of the triangle is
(1) 64 3 (2) 48 3 (3) 192 3 (4) none of these
Q.27 Radius of the circum-circle of the triangle is(1) 4 (2) 8 (3) 16 (4) 32
Paragraph for Question no. 28 to 30Conic possesses enormous properties which can be probed by taking standard forms. Unlike circlethese properties rarely follow by geometrical considerations. Most of the properties of conics are provedanalytically. For example, the properties of a parabola can be proved by taking its standard equationy2 = 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 andGrespectively, 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, thenKSP 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 1
21 at2,at and B
222 at2,at .
Column-I Column-II
(1) AB is a normal chord if (P) t2
= – t1
–2
1
(2) AB is a focal chord (Q) t2
=1t
4
(3) AB subtends 90° at point (0, 0) if (R) t2
=1t
1
(S) t2
= – t1
–1t
2
Q.32 Column-I Column-II(1) The focus of the parabola (y – 4)2 = 12 (x – 2) (P) (1, 2)(2) The vertex of the parabola y2 – 5x – 4y + 9 = 0 (Q) (–3, 1)(3) The foot of the directrix of the parabola x2 + 8y = 0 (R) (3, 0)(4) The vertex of the parabola x2 + 6x – 2y + 11 = 0 (S) (5, 4)
(T) (2, 0)
Q.33 Column-I Column-II(1) The directrix of the parabola y2 – 2y + 8x – 23 = 0 (P) x + 2 = 0(2) The equation to the latus rectum of x2 – 2x – 4y – 3 = 0 (Q) y – 5 = 0(3) The axis of the parabola x2 + 8x + 12y + 4 = 0 is (R) x + 4 = 0(4) Equation to the tangent at vertex of y2 – 6y – 12x – 15 = 0 (S) x – 5 = 0
(T) y – 2 = 0
Q.34 Column-I Column-II
(1) The length of the latus rectum of the parabola whose (P)2
25
focus is (4, 5) and the vertex (3, 6)(2) The line 2x – y + 2 = 0 touches the parabola y2 = 4px. Then p = (Q) 4(3) The length of the focal chord of y2 = 8x drawn through (8, 8) is (R) – 2(4) y = 3x + c is a tangent to y2 = 12x then c = (S) 1
(T) 4 2
ANSWER KEY
EXERCISE-1Q.1 1 Q.2 3 Q.3 2 Q.4 3 Q.5 2
Q.6 3 Q.7 1 Q.8 4 Q.9 3 Q.10 1
Q.11 3 Q.12 3 Q.13 2 Q.14 4 Q.15 3
Q.16 3 Q.17 3 Q.18 2 Q.19 4 Q.20 2
Q.21 2 Q.22 1 Q.23 3 Q.24 4 Q.25 2
Q.26 2 Q.27 1 Q.28 2 Q.29 3 Q.30 3
EXERCISE-2
Q.1 2 Q.2 2 Q.3 1 Q.4 2 Q.5 3Q.6 1 Q.7 2 Q.8 2 Q.9 3 Q.10 2Q.11 2 Q.12 1 Q.13 1 Q.14 3 Q.15 1
Q.16 1 Q.17 4 Q.18 3 Q.19 2 Q.20 2
Q.21 1 Q.22 2 Q.23 4 Q.24 3 Q.25 4Q.26 1 Q.27 2 Q.28 3 Q.29 1 Q.30 1
EXERCISE-3Q.1 2 Q.2 1 Q.3 1 Q.4 3 Q.5 3Q.6 1 Q.7 3 Q.8 3 Q.9 3 Q.10 2Q.11 3 Q.12 4 Q.13 1 Q.14 1 Q.15 3
Q.16 1 Q.17 23 Q.18 2 Q.19 2 Q.20 2
Q.21 1 Q.22 3 Q.23 3 Q.24 4Q.25 3
EXERCISE-4Q.1 1 Q.2 2 Q.3 3 Q.4 3 Q.5 1
Q.6 1 Q.7 2 Q.8 1 Q.9 1 Q.10 1
Q.11 1 Q.12 3 Q.13 1 Q.14 1 Q.15 1
Q.16 4 Q.17 4 Q.18 3 Q.19 2 Q.20 3
Q.21 1 Q.22 2 Q.23 4 Q.24 3 Q.25 2
Q.26 3 Q.27 3 Q.28 2 Q.29 3 Q.30 3
Q.31 (1) S ; (2) R ; (3) Q Q.32 (1) S ; (2) P ; (3) Q ; (4) T
Q.33 (1) S ; (2) T ; (3) R ; (4) P Q.34 (1) T ; (2) Q ; (3) P ; (4) S
