JEE Mathematics Ellipse and Hyperbola MCQs Set 02

Choose the most appropriate option (a, b, c or d)

Question. The number of values of m for which the line \( y = mx + \sqrt{m^2 - 4} \) touches the hyperbola \( 4(x^2 – 1) = y^2 \) is
(a) two
(b) zero
(c) one
(d) infinite
Answer: (d) infinite

Question. The value of c for which the line \( y = 3x + c \) touches the ellipse \( 16x^2 + y^2 = 16 \) is
(a) 5
(b) 1
(c) 4
(d) 3
Answer: (a) 5

Question. The number of values of \( \phi \in [0, 2\pi] \) for which the line \( 2x \cos \phi + 3y \sin \phi = 6 \) touches the ellipse \( 4x^2 + 9y^2 = 36 \) is
(a) four
(b) two
(c) one
(d) infinite
Answer: (d) infinite

Question. The line \( 3x + 5y = k \) is a tangent to the ellipse \( 16x^2 + 25y^2 = 400 \) if k is
(a) \( \pm 5 \)
(b) \( \pm 15 \)
(c) \( \pm 25 \)
(d) \( \pm 10 \)
Answer: (c) \( \pm 25 \)

Question. The line \( px + qy = r \) touches the hyperbola \( b^2x^2 – a^2y^2 = a^2b^2 \) if
(a) \( a^2p^2 + b^2q^2 = r^2 \)
(b) \( a^2p^2 – b^2q^2 = r^2 \)
(c) \( a^2q^2 + b^2p^2 = r^2 \)
(d) \( a^2q^2 – b^2p^2 = r^2 \)
Answer: (b) \( a^2p^2 – b^2q^2 = r^2 \)

Question. The equation of the tangent to the ellipse \( \frac{x^2}{25} + \frac{y^2}{16} = 1 \), which is parallel to the line \( y = 3x \), is
(a) \( y = 3x \pm \sqrt{241} \)
(b) \( y = 3x + 13 \)
(c) \( y = 3x + \sqrt{209} \)
(d) none of the options
Answer: (a) \( y = 3x \pm \sqrt{241} \)

Question. The equation of the tangent to the hyperbola \( x^2 – 2y^2 = 18 \), which is perpendicular to the line \( x – y = 0 \), is
(a) \( x + y = 3 \)
(b) \( x + y + 2 = 0 \)
(c) \( x + y = 3\sqrt{2} \)
(d) \( x + y + 3\sqrt{2} = 0 \)
Answer: (a) \( x + y = 3 \)

Question. If the tangents from the point \( (\lambda, 3) \) to the ellipse \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \) are at right angles then \( \lambda \) is
(a) \( \pm 1 \)
(b) \( \pm 3 \)
(c) \( \pm 2 \)
(d) none of the options
Answer: (c) \( \pm 2 \)

Question. A point on the ellipse \( x^2 + 3y^2 = 9 \), where the tangent is parallel to the line \( y – x = 0 \), is
(a) \( (\sqrt{3}, \sqrt{2}) \)
(b) \( \left( -\frac{3\sqrt{3}}{2}, -\frac{\sqrt{3}}{2} \right) \)
(c) \( \left( -\frac{3\sqrt{3}}{2}, \frac{\sqrt{3}}{2} \right) \)
(d) \( (-\sqrt{3}, \sqrt{2}) \)
Answer: (c) \( \left( -\frac{3\sqrt{3}}{2}, \frac{\sqrt{3}}{2} \right) \)

Question. The ordinate of the point of contact of a tangent is 2. Then the equation of the tangent to \( x^2 + 4y^2 = 25 \) is
(a) \( 3x + 8y = 25 \)
(b) \( 8x + 3y = 25 \)
(c) \( 3x – 8y = 25 \)
(d) none of the options
Answer: (a) \( 3x + 8y = 25 \)

Question. The tangent to the ellipse \( 16x^2 + 9y^2 = 144 \), making equal intercepts on both the axes, is
(a) \( y = x + 3 \)
(b) \( y = x - 2 \)
(c) \( x + y = 5 \)
(d) \( y = -x + 4 \)
Answer: (c) \( x + y = 5 \)

Question. If the tangent to the ellipse \( x^2 + 4y^2 = 16 \) at the point ‘\( \phi \)’ is a normal to the circle \( x^2 + y^2 – 8x – 4y = 0 \) then \( \phi \) is equal to
(a) \( \frac{\pi}{2} \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{3} \)
(d) \( -\frac{\pi}{4} \)
Answer: (a) \( \frac{\pi}{2} \)

Question. The area of the quadrilateral formed by tangents at the end points of latus recta of the ellipse \( \frac{x^2}{9} + \frac{y^2}{5} = 1 \) is
(a) \( \frac{27}{4} \) unit²
(b) 9 unit²
(c) \( \frac{27}{2} \) unit²
(d) 27 unit²
Answer: (d) 27 unit²

Question. The tangent at \( (3\sqrt{3}\cos \theta, \sin \theta) \) is drawn to the ellipse \( \frac{x^2}{27} + y^2 = 1 \). Then the value of \( \theta \) such that the sum of intercepts on axes made by the tangent is minimum is
(a) \( \frac{\pi}{3} \)
(b) \( \frac{\pi}{6} \)
(c) \( \frac{\pi}{8} \)
(d) \( \frac{\pi}{4} \)
Answer: (b) \( \frac{\pi}{6} \)

Question. The number of normals that can be drawn to the curve \( 4x^2 + 9y^2 = 36 \) from an external point, in general, is
(a) 1
(b) 3
(c) 4
(d) infinite
Answer: (c) 4

Question. The number of distinct normal lines from the exterior point (0, c), c > b, to the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is
(a) 3
(b) 4
(c) 2
(d) 1
Answer: (d) 1

Question. The equation of the normal to the ellipse \( x^2 + 4y^2 = 16 \) at the end of the latus rectum in the first quadrant is
(a) \( 2x + \sqrt{3}(y + 3) = 0 \)
(b) \( 2x = \sqrt{3}(y + 3) \)
(c) \( \sqrt{3}x = 2(y + 3) \)
(d) none of the options
Answer: (b) \( 2x = \sqrt{3}(y + 3) \)

Question. If the tangent and the normal to \( x^2 – y^2 = 4 \) at a point cut off intercepts \( a_1, a_2 \) on the x-axis respectively and \( b_1, b_2 \) on the y-axis respectively then the value of \( a_1a_2 + b_1b_2 \) is
(a) 1
(b) -1
(c) 0
(d) 4
Answer: (c) 0

Question. The normal to the rectangular hyperbola \( xy = c^2 \) at the point ‘\( t_1 \)’ meets the curve again at the point ‘\( t_2 \)’. The value of \( t_1^3 \cdot t_2 \) is
(a) 1
(b) c
(c) –c
(d) -1
Answer: (d) -1

Question. If P and Q are the ends of a pair of conjugate diameter and C is the centre of the ellipse \( 4x^2 + 9y^2 = 36 \) then the area of the \( \Delta CPQ \) is
(a) 6 unit²
(b) 3 unit²
(c) 2 unit²
(d) 12 unit²
Answer: (b) 3 unit²

Question. If \( y = x \) and \( 3y + 2x = 0 \) are the equations of a pair of conjugate diameters of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) then its eccentricity is
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{1}{\sqrt{3}} \)
(d) \( \frac{\sqrt{3}}{2} \)
Answer: (c) \( \frac{1}{\sqrt{3}} \)

Question. The locus of a point \( P(\alpha, \beta) \) moving under the condition that the line \( y = \alpha x + \beta \) is a tangent to the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) is
(a) a circle
(b) an ellipse
(c) a hyperbola
(d) a parabola
Answer: (c) a hyperbola

Choose the correct option. One or more option may be correct.

Question. A focus of the hyperbola \( 25x^2 – 36y^2 = 225 \) is
(a) \( (\sqrt{61}, 0) \)
(b) \( (\frac{1}{2}\sqrt{61}, 0) \)
(c) \( (-\sqrt{61}, 0) \)
(d) \( (-\frac{1}{2}\sqrt{6, 0}) \)
Answer: (b) \( (\frac{1}{2}\sqrt{61}, 0) \) and (d) \( (-\frac{1}{2}\sqrt{61, 0}) \)

Question. The point P on the ellipse \( 4x^2 + 9y^2 = 36 \) is such that the area of the \( \Delta PF_1F_2 = \sqrt{10} \) where \( F_1, F_2 \) are foci. Then P has the coordinates
(a) \( (\frac{3}{\sqrt{2}}, \sqrt{2}) \)
(b) \( (\frac{3}{2}, 2) \)
(c) \( (-\frac{3}{2}, -2) \)
(d) \( (-\frac{3}{\sqrt{2}}, -\sqrt{2}) \)
Answer: (a) \( (\frac{3}{\sqrt{2}}, \sqrt{2}) \) and (d) \( (-\frac{3}{\sqrt{2}}, -\sqrt{2}) \)

Question. A point on the ellipse \( x^2 + 3y^2 = 37 \), where the normal is parallel to the line \( 6x – 5y = 2 \), is
(a) (5, -2)
(b) (5, 2)
(c) (-5, 2)
(d) (-5, -2)
Answer: (b) (5, 2) and (d) (-5, -2)