JEE Mathematics Hyperbola MCQs Set 02

Question:   The eccentricity of the conic x² – y² – 4x + 4y + 16 = 0 is

• a) √2
• b) 1
• c) 2
• d) 1/2

Answer: √2

Question: The equation 9x² – 16y² – 18x + 32y – 151 = 0 represent a hyperbola -

• a) Equation of whose directrix is x = 21/5 and x = -11/5
• b) None of these
• c) Length of whose transverse axes is 4
• d) Length of whose latusrectum is 9

Answer: Equation of whose directrix is x = 21/5 and x = -11/5

Question: For what value of λ does the line y = 2x + λ touches the hyperbola 16x² – 9y2 = 144 ?

• a) ± 2 √5
• b) ± 3 √5
• c) ± 4 √5
• d) None of these

Answer: ± 2 √5

Question: Find the equation of the tangent to the hyperbola x² – 4y² = 36 which is perpendicular to the line x – y + 4 = 0.

• a) x + y ± 3 √3 = 0
• b) x + y ± 2 √3 = 0
• c) x + y ± 5 √3 = 0
• d) x – y ± 3 √3 = 0

Answer: x + y ± 3 √3 = 0

Question: If the normals at (x_i , y_i ) i = 1, 2, 3, 4 to the rectangular hyperbola xy = 2 meet at the point (3, 4), then

(1) x₁ + x₂ + x₃ + x₄ = 3 (2) y₁ + y₂ + y₃ + y₄ = 4
(3) x₁ x₂ x₃ x₄ = –4        (4) y₁ y₂ y3 y₄ = 4

• a) 1, 2 and 3 are correct
• b) 2 and 4 are correct
• c) 1 and 2 are correct
• d) 1 and 3 are correct

Answer: 1, 2 and 3 are correct

Question: If the circle x₂ + y₂ = 1 cuts the rectangular hyperbola xy = 1 in four points ( x_i , y_i ); i = 1, 2, 3, 4 then.

(1) x₁x₂ x₃x₄ = – 1          (2) y₁y₂ y₃ y₄ = 1
(3) x₁ + x₂ + x₃ + x₄ = 1 (4) y₁ + y₂ + y₃ + y₄ = 0

• a) 2 and 4 are correct
• b) 1 and 2 are correct
• c) 1, 2 and 3 are correct
• d) 1 and 3 are correct

Answer: 2 and 4 are correct

Question:  

Statement-1 : There can be infinite points from where we can draw two mutually perpendicular tangents to the hyperbola x²/9 - y²/16 =1

atement-2 : The director circle in case of hyperbola x²/9 - y²/16 =1will not exist because a² < b² and director circle is x² + y² = a² – b².

• a) Statement-1 is False, Statement-2 is True.
• b) Statement-1 is True, Statement-2 is False.
• c) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
• d) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Answer: Statement-1 is False, Statement-2 is True.

Question:

Statement-1 : With respect to a hyperbola x²/9 - y²/16 =1 perpendicular are drawn from a point (5, 0) on the lines 3y ± 4x = 0, then their feet lie on circle x² + y² = 16.

Statement-2 : If from any foci of a hyperbola perpendicular are drawn on the asymptotes of the hyperbola then their feet lie on auxiliary circle.

• a) Statement-1 is False, Statement-2 is True.
• b) Statement-1 is True, Statement-2 is False.
• c) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
• d) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Answer: Statement-1 is False, Statement-2 is True.

Question: Two straight lines pass through the fixed points (± a,0) and have gradients whose product is k, then the locus of the points of inter-section of the lines is

• a) hyperbola
• b) parabola
• c) circle
• d) None of these

Answer: hyperbola

Question: The eccentricity of the conic x² – y² – 4x + 4y + 16 = 0 is

• a) √2
• b) 1/2
• c) 2

Answer: √2

Question: The equation 16x² = 3y² – 32x + 12y – 44= 0 represents a hyperbola

• a) whose eccentricity is √19/3
• b) whose centre is (– 1, 2)
• c) the length of whose conjugate axis is 4
• d) the length of whose transverse axis is 4√3

Answer: whose eccentricity is √19/3

Question: The line 5x + 12y = 9 touches the hyperbola x² – 9y² = 9 at the point

• a) (5, – 4/3)
• b) None of these
• c) (– 5,4/3)
• d) (3, – 1/2)

Answer: (5, – 4/3)

Question: Locus of the mid points of the chords of the circle x² + y² = 16, which are tangent to the hyperbola 9x² – 16y² = 144 is

• a) (x² + y²)² = 16x² – 9y²
• b) (x² + y²)² = 5x² – 4y²
• c) (x² + y²)2 = 19x² – 8y²
• d) (x² + y²)² = 18x² – 8y²

Answer: (x² + y²)² = 16x² – 9y²

Question: If e and e' be the eccentricities of a hyperbola and its conjugate then the value of 1/e² + 1/e^{'  2} =

• a) 1
• b) 4
• c) 0
• d) 2

Answer: 1

Question: The equation of the common tangents to the parabola y² = 8x and the hyperbola 3x² – y² = 3 is-

• a) 2x ± y + 1 = 0
• b) x ± 2y + 1 = 0
• c) x ± y + 1 = 0
• d) x ± y + 2 = 0

Answer: 2x ± y + 1 = 0

Question: The locus of the point of intersection of the lines √3 x –y– 4 √3 k =0 and √3 kx + ky – 4 √3 = 0 for different values of k is-

• a) Hyperbola
• b) Parabola
• c) Ellipse
• d) Circle

Answer: Hyperbola

Question: The area of a triangle formed by the lines x – y = 0, x + y = 0 and any tangent to the hyperbola x² – y² = a² is-

• a) a²
• b) 3a²
• c) 2a²
• d) 4a²

Answer: a²

Question: Find the equation of the hyperbola whose directrix is 2x + y = 1, focus (1,2) and eccentricity √3 .

• a) 7x² – 2y² + 12xy – 2x + 14y – 22 = 0
• b) 7x² – 2y² – 12xy – 2x + 14y – 18 = 0
• c) 7x² + 2y² – 12xy – 2x + 14y – 22 = 0
• d) 7x² + 2y² + 12xy – 2x + 14y + 11 = 0

Answer: 7x² – 2y² + 12xy – 2x + 14y – 22 = 0

Question: From the points on the circle x² + y² = a², tangents are drawn to the hyperbola x² – y² = a²; then the locus of the middle points of the chords of contact is the curve

• a) (x² – y²)² = a²(x² + y²)
• b) (x² – y²)² = a²(x² – y²)
• c) (x² + y²)² = a²(x² + y²)
• d) (x² – y²)² = 2a²(x² + y²)

Answer: (x² – y²)² = a²(x² + y²)

Question: Coordinates of foci, eccentricity of the hyperbola 9x² – 16y² – 72x + 96y– 144 = 0 are

• a) (9,3), (– 1,3), 5/4
• b) (3,3), (– 1,3), 1/4
• c) (1,3), (– 2,3), 5/4
• d) (2,3), (–2, 3), 3/4

Answer: (9,3), (– 1,3), 5/4

Question: The locus of the mid point of the chords of the circle x² + y² = 16, which are tangent to the hyperbola 9x² – 16y² = 144 is-

• a) (x² + y²)² = a²x² – b²y²
• b) x² + y² = a² – b²
• c) (x² + y²)² = a² – b²
• d) (x² + y²)² = a² + b²

Answer: (x² + y²)² = a²x² – b²y²