Class 11 Mathematics Conic Sections MCQs Set H

Question: If the equation of parabola is x² = -9y, then equation of directrix and length of latusrectum are
(a) y = −9/4,8
(b) x =9/4,9
(c) y = 9/4,9 
(d) None of these 
Answer: c

Question: If the vertex of a parabola is the point( -3,0 ) and the directrix is the line x +5 =0, then equation of parabola is
(a) y² = 8(x+3)
(b) x2 = 8(y+3)
(c) y2 = -8(8+3)
(d) y2 = 8(x+5) 
Answer: a

Question: The parametric coordinates of any point on the parabola y² = 4ax can be
(a) ( a-at²,-2at 
(b) (a− at²,2at)
(c) (a sin² t,-2a sin t) 
(d) (a sint,-2 a cot) 
Answer: c

Question: A circle touches the y-axis at the point (0, 4) and passes through the point (2, 0). Which of the following lines is not a tangent to this circle?
(a) 4x – 3y + 17 = 0
(b) 3x – 4y – 24 = 0
(c) 3x + 4y – 6 = 0
(d) 4x + 3y – 8 = 0
Answer: d

Question: If a circle passing through the point (–1, 0) touches y-axis at (0, 2), then the length of the chord of the circle along the x-axis is :
(a) 3/2
(b) 3
(c) 5/2
(d) 5
Answer: b

Question: The equation (X-2)² + (Y-3)²=(3X+4Y-2/5)² represents
(a) a parabola
(b) a pair of straight lines
(c) an ellipse
(d) a hyperbola 
Answer: a

Question: A line drawn through the point P(4, 7) cuts the circle x² + y² = 9 at the points A and B. Then PA·PB is equal to :
(a) 53
(b) 56
(c) 74
(d) 65
Answer: b

Question: The focus of the parabola y²=4y-4x is
(a) (0, 2)
(b) (1, 2)
(c) (2, 0)
(d) (2, 1) 
Answer: a

Question: The equation of tangent to the parabola y²=9x which goes through the point (4, 10) is
(a) x+ 4y +1=0
(b) 9x + 4y + 4=0 
(c) x- 4y+ 36 = 0
(d) 9x + 4y+9=0 
Answer: c

Question: The tangent to the parabola y²=4ax at the point(a,2a) makes with x-axis an angle equal to
(a) π/3
(b) p/4
(c) p/2
(d) p/6 
Answer: b

Question: Two circles with equal radii are intersecting at the points (0, 1) and (0, –1). The tangent at the point (0, 1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is :
(a) 1
(b) 2
(c) 2√2
(d) √2
Answer: d

Question: The equation represents λX²+4XY+Y²+λx+3Y+2=0 represents a parabola, if λis
(a) -4
(b) 4
(c) 0
(d) None of these 
Answer: b

Question: Two sets A and B are as under :
A = {(a, b) ∈R × R : | a -5| <1 and | b – 5 | < 1};
B = {(a,b) ∈R × R : 4(a- 6)² + 9(b- 5)² ≤ 36}. Then :
(a) A ⊂ B
(b) A ∩ B = Φ (an empty set)
(c) neither A ⊂ B nor B ⊂ A
(d) B ⊂ A
Answer: a

Question: If the curves, x² – 6x + y² + 8 = 0 and x² – 8y + y² + 16 – k = 0, (k > 0) touch each other at a point, then the largest value of k is ______.
Answer: 36

Question: The number of integral values of k for which the line, 3x + 4y = k intersects the circle, x² + y² – 2x – 4y + 4 = 0 at two distinct points is ________.
Answer: 9

ASSERTION- REASON TYPE QUESTIONS

(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion:
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Question: Assertion : Centre of the circle x² + y² – 6x + 4y – 12 = 0 is (3, –2).
Reason : The coordinates of the centre of the circle x² + y² + 2gx + 2fy + c = 0 are (–1/2 coefficient of x, – 1/2 coefficient of y)
Answer: a

Question: Assertion : Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola.
Reason : The equation of a hyperbola with foci on the y-axis is : x²/a² − y²/b²= 1
Answer: c

Question: Assertion : Ellipse x²/25 + y²/16= 1 and 12x² – 4y² = 27 intersect each other at right angle.
Reason : Whenever focal conics intersect, they intersect each other orthogonally.
Answer: a

Question: Parabola is symmetric with respect to the axis of the parabola.
Assertion: If the equation has a term y² , then the axis of symmetry is along the x-axis.
Reason: If the eqution has a term x², then the axis of symmetry is along the x-axis.
Answer: c

Question. The equation of an ellipse whose focus is (–1, 1), whose directrix is x − y + 3 = 0 and whose eccentricity is 1/2 is given by:
(a) 7x² + 2xy + 7y² +10x −10y + 7 = 0
(b) 7x² − 2xy + 7 y² −10x +10y + 7 = 0
(c) 7x² − 2xy + 7y² −10x −10y − 7 = 0
(d) 7x² − 2xy + 7 y² +10x +10y − 7 = 0

Answer: A

Question. The equation of parabola whose focus is (5, 3) and directrix is 3x − 4y +1= 0, is:
(a) (4x + 3y)² − 256x −142y + 849 = 0
(b) (4x − 3y)² − 256x −142y + 849 = 0
(c) (3x + 4y)² −142x − 256y + 849 = 0
(d) (3x − 4y)² − 256x −142y + 849 = 0

Answer: A

Question. If the parabola 2 y = 4ax passes through (–3, 2), then length of its latus rectum is:
(a) 2/3
(b) 1/3
(c) 4/3
(d) 4

Answer: C

Question. The number of values of ‘c’ such that the straight line y = 4x + c touches the curve x²/4 + y² =1is :
(a) 0
(b) 1
(c) 2
(d) Infinite

Answer: C

Question. If the normal at any point P on the ellipse cuts the major and minor axes in G and g respectively and C be the centre of the ellipse, then:
(a) a² (CG)² + b² (Cg) = (a² − b² )²
(b) a² (CG)² − b² (Cg) = (a² − b² )²
(c) a² (CG)² − b² (Cg) = (a² + b² )²
(d) None of these

Answer:

Question. x − 2 = t² , y = 2t are the parametric equations of the parabola:
(a) y² = 4x
(b) y² = −4x
(c) x²  = −4y
(d) y²  = 4(x − 2)

Answer: D

Question. The straight line y = 2x +λ does not meet the parabola y² = 2x, if:
(a) λ <1/4
(b) λ >1/4
(c) λ = 4
(d) λ =1

Answer: B

Question. If x + y = k is a normal to the parabola y² = 12x, then k is:
(a) 3
(b) 9
(c) –9
(d) –3

Answer: B

Question. The normals at three points P, Q, R of the parabola y² = 4ax meet in (h, k), the centroid of triangle PQR lies on:
(a) x = 0
(b) y = 0
(c) x = −a
(d) y = a

Answer: B

Question. If m₁ and m₂ are the slopes of the tangents to the hyperbola x²/25 – y²/16 = 1 which pass through the point (6,2), then:     
(a) m₁+m₂ = 24/11
(b) m₁.m₂ = 20/11
(c) m₁+m₂ = 48/11
(d) m₁.m₂ = 11/20

Answer: A,B

Question. If the points (au² ,2au) and (av² ,2av) are the extremities of a focal chord of the parabola y² = 4ax, then:
(a) uv −1 = 0
(b) uv +1 = 0
(c) u + v = 0
(d) u −v = 0

Answer: B

Question. Equation of diameter of parabola y² = x corresponding to the chord x − y +1 = 0 is:
(a) 2y = 3
(b) 2y =1
(c) 2y = 5
(d) y = 1

Answer: B

Question. The length of the sub-tangent to the parabola y² = 16x at the point, whose abscissa is 4, is:
(a) 2
(b) 4
(c) 8
(d) None of these

Answer: C

Question. The pole of the line 2x = y with respect to the parabola y² = 2x is:
(a) (0 , 1/2)
(b) (1/2 , 0) 
(c) (0 , –1/2)
(d) None of these 

Answer: A

Question. A ray of light moving parallel to the x-axis gets reflected from a parabolic mirror whose equation is ( y − 2)² = 4(x +1). after reflection, the ray must pass through the point:
(a) (0, 2)
(b) (2, 0)
(c) (0, –2)
(d) (–1, 2)

Answer: A

Question. If the normal at (ct , c/t) on the curve xy = c² meets the curve again in t′, then:
(a) t' = -1/t₃
(b)  t' = -1/t
(c)  t' = -1/t₂
(d)  t₂ = -1/t₂

Answer: A

Question. If a circle cuts a rectangular hyperbola xy² = c in A, B, C, D and the parameters of these four points be t₁ , t₂ , t₃ and t₄ respectively. Then:
(a) t₁t₂ = t₃t₄
(b)  t₁t₂t₃t₄ = 1
(c)  t₁ = t₂ 
(d)  t₃ = t₄

Answer: B

Question. If 1 P(x, y),F = (3,0), 2 F = (−3,0) and 16x² + 25y² = 400, then PF₁ + PF₂ equals:
(a) 8
(b) 6
(c) 10
(d) 12

Answer: C

Question. The equation x² − 2xy + y² + 3x + 2 = 0 represents: 
(a) A parabola
(b) An ellipse
(c) A hyperbola
(d) A circle

Answer: A

Question. The equation of a directrix of the ellipse x²/16 + y²/25 = 1 is :
(a) y = 25/3
(b) x = 3
(c) x = −3
(d) x = 3/25

Answer: A

Question. If the tangent to the parabola y² = ax makes an angle of 45º with x-axis, then the point of contact is:
(a) (a/2 , a/2)
(b) (a/4 , a/4)
(c) (a/2 , a/4)
(d) (a/4 , a/2)

Answer: D

Question. The line x − y + 2 = 0 touches the parabola y² = 8x at the point:
(a) (2,–4)
(b) (1,2 √2)
(c) (4,−4 √2)
(d) (2, 4)

Answer: D

Question. The distance of the point 'θ ' on the ellipse x²/9 + y²/4 = 1 from a focus is:
(a) a(e + cosθ )
(b) a(e − cosθ )
(c) a(1+ e cosθ )
(d) a(1+ 2ecosθ )

Answer: C

Question. The centre of 14x² − 4xy +11y² − 44x − 58y + 71 = 0 is:
(a) (2, 3)
(b) (2, –3)
(c) (–2, 3)
(d) (–2, –3)

Answer: A

Question. Let E be the ellipse x²/9 + y²/4 = 1 and C be the circle x² + y² = 9. Let P and Q be the points (1, 2) and (2, 1) respectively. Then:
(a) Q lies inside C but outside E
(b) Q lies outside both C and E
(c) P lies inside both C and E
(d) P lies inside C but outside E

Answer: D

Question. What will be the equation of the chord of contact of tangents drawn from (3, 2) to the ellipse x² + 4y² = 9 ?
(a) 3x + 8y = 9
(b) 3x + 8y = 25
(c) 3x + 4y = 9
(d) 3x + 8y + 9 = 0

Answer: A

Question. The pole of the straight line x + 4y = 4 with respect to ellipse x² + 4y² = 4 is:
(a) (1, 4)
(b) (1, 1)
(c) (4, 1)
(d) (4, 4)

Answer: B

Question. If one end of a diameter of the ellipse 4x² + y² = 16 is ( √3, 2), then the other end is:
(a) (− √3, 2)
(b) ( √3,− 2)
(c) (− √3,− 2)
(d) (0,0)

Answer: C

Question. If θ and φ are eccentric angles of the ends of a pair of conjugate diameters of the ellipse x²/a² + y²/b² = 1,then θ −φ is equal to:
(a) ± π/2
(b) ± π
(c) 0
(d) None of these

Answer: A

Question. In the ellipse 25x² + 9y² + 150x – 190y + 225 = 0 ?       
(a) foci are at (3,1), (3, 9)
(b) e = 4/5
(c) centre is (5,3)
(d) major axis is

Answer:   A,B

Question. Length of sub-tangent and subnormal at the point (Image 30)       

Answer: A

Question. The equation of a parabola is 2 y = 4x. P(1,3) and Q(1,1) are two points in the xy-plane. Then, for the parabola:
(a) P and Q are exterior points
(b) P is an interior point while Q is an exterior point
(c) P and Q are interior points
(d) P is an exterior point while Q is an interior point

Answer: D

Question. The equation of the conic with focus at (1, – 1), directrix along x − y +1 = 0 and with eccentricity 2 is:
(a) x² − y² =1
(b) xy =1
(c) 2xy + 4x − 4y −1 = 0
(d) 2xy + 4x − 4y −1 = 0

Answer: C

Question. The locus of the point of intersection of tangents to the hyperbola 4x² − 9y² = 36 which meet at a constant angle π / 4, is:
(a) (x² + y² − 5)² = 4(9y² − 4x² + 36)
(b) (x² + y² − 5)² = 4(9y² − 4x² + 36)
(c) 4(x² + y² − 5)² = (9y² − 4x² + 36)
(d) None of these

Answer: A

Question. The equation of the normal to the hyperbola  x²/16 – y²/9 = 1 at the point (8,3√3) is
(a) √3x + 2y = 25
(b) x + y = 25
(c) y + 2x = 25
(d) 2x + √3y = 25

Answer: D

Question. If the normal at 'φ ' on the hyperbola x²/a² + y²/b² = 1 meets transverse axis at G, then AG.A'G = ? (Where A and A' are the vertices of the hyperbola)
(a) a² (e4 sec2 φ −1)
(b) (a2 e4 sec2 φ −1)
(c) a2 (1− e4 sec2 φ )
(d) None of these

Answer: A

Question. The equation of the chord of contact of tangents drawn from a point (2, –1) to the hyperbola 16x² − 9y² =144 is:
(a) 32x + 9y = 144
(b) 32x + 9y = 55
(c) 32x + 9y +144 = 0
(d) 32x + 9y + 55 = 0

Answer: A

Question. The point of intersection of tangents drawn to the hyperbola x²/a² + y²/b² = 1 at the points where it is intersected by the line lx + my + n = 0 is:  (Image 38)         

Answer: A

Question. If the polar of a point w.r.t. x²/a² + y²/b² = 1 touches the hyperbola x²/a² + y²/b² = 1 then the locus of the point is:
(a) Given hyperbola
(b) Ellipse
(c) Circle
(d) None of these

Answer: A

Question. If a pair of conjugate diameters meet the hyperbola and its conjugate in P and D respectively, then CP² −CD² = ?
(a) a² + b²
(b) a² − b²
(c) a²/b²
(d) None of these

Answer: B

Question. If the tangent at the point (asecα , b tanα ) to the hyperbola x²/a² - y²/b² = meets the transverse axis at T, then the distance of T form a focus of the hyperbola is:         
(a) a(e − cosα )
(b) b(e + cosα )
(c) a(e + cosα )
(d) √(a² e² + b² + cot² α)

Answer: A,C

Question. From any point on the hyperbola,  x²/a² + y²/b² = 1 tangents are drawn to the hyperbola x²/a² + y²/b² = 2 The area cut-off by the chord of contact on the asymptotes is equal to:
(a) ab/2
(b) ab
(c) 2ab
(d) 4ab

Answer: D

Question. The area of the quadrilateral formed by the tangents at the end points of latus- rectum to the ellipse x²/9 + y²/5 = 1 is :
(a) 27/4 sq. units
(b) 9 sq. units
(c) 27/2 sq. units
(d) 27sq. units

Answer: D

Question. The equation of normal at the point (0, 3) of the ellipse 9x² + 5y² = 45 is:
(a) y −3 = 0
(b) y + 3 = 0
(c) x-axis
(d) y-axis

Answer: D

Question. The number of tangents to the hyperbola x²/4 – y²/3 = 1 through (4, 1) is:
(a) 1
(b) 2
(c) 0
(d) 3

Answer: C

Question. The points of contact of the line i y = x −1 with 3x² − 4y² = 12 is:
(a) (4, 3)
(b) (3, 4)
(c) (4,–3)
(d) None of these

Answer: A

Question. The combined equation of the asymptotes of the hyperbola 2x² + 5xy + 2y² + 4x + 5y = 0 ?
(a) 2x² + 5xy + 2y² = 0
(b) 2x² + 5xy + 2y² − 4x + 5y + 2 = 0 = 0
(c) 2x² + 5xy + 2y² + 4x + 5y − 2 = 0
(d) 2x² + 5xy + 2y² + 4x + 5y + 2 = 0

Answer: D

Question. If 5x² +λy² = 20 represents a rectangular hyperbola, then λ equals:
(a) 5
(b) 4
(c) – 5
(d) None of these

Answer: C

Question. Equation of common tangent of y = x², y = – x² + 4x – 4 is:                    
(a) y = 4 (x – 1)
(b) y = 0
(c) y = – 4 (x – 1)
(d) y = – 30 x – 50

Answer: A,B

Question. Let P(x₁,y₁) and Q(x₂,y₂), y₁ < 0, y₂ < 0, be the end points of the latus rectum of the ellipse x² + 4y² = 4.The equations of parabolas with latus rectum PQ are:         
(a) x² + 2√3y = 3+ √3
(b)x² − 2√3y = 3+ √3
(c) x² + 2√3y = 3− √3
(d) x² − 2√3y = 3− √3

Answer: B,C