Hyperbola-JEE-Mains-PYQ’s

Mathematics Questions

Let the ellipse $E : \frac{x^2}{144} + \frac{y^2}{169} = 1$ and the hyperbola $H : \frac{x^2}{16} - \frac{y^2}{\lambda^2} = -1$ have the same foci. If $e$ and $L$ respectively denote the eccentricity and the length of the latus rectum of $H$ , then the value of $24(e + L)$ is :

(A) 296

(B) 126

(D) 148

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Let $PQ$ be a chord of the hyperbola $\frac{x^2}{4} - \frac{y^2}{b^2} = 1$ , perpendicular to the x-axis such that $OPQ$ is an equilateral triangle, $O$ being the centre of the hyperbola. If the eccentricity of the hyperbola is $\sqrt{3}$ , then the area of the triangle $OPQ$ is :

(A) $2\sqrt{3}$

(B) $\frac{11}{5}$

(D) $\frac{9}{5}$

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Let the domain of the function $f(x) = \log_3 \log_5 \log_7 (9x - x^2 - 13)$ be the interval $(m, n)$ . Let the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ have eccentricity $\frac{n}{3}$ and the length of the latus rectum $\frac{8m}{3}$ . Then $b^2 - a^2$ is equal to :

(A) 7

(B) 9

(D) 5

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Let $P(10, 2\sqrt{15})$ be a point on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ , whose foci are $S$ and $S'$ . If the length of its latus rectum is 8, then the square of the area of $\Delta PSS'$ is equal to :

(A) 4200

(B) 1462

(D) 2700

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If the line $\alpha x + 2y = 1$ , where $\alpha \in \mathbb{R}$ , does not meet the hyperbola $x^2 - 9y^2 = 9$ , then a possible value of $\alpha$ is :

(A) 0.6

(B) 0.7

(D) 0.5

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Let the foci of a hyperbola coincide with the foci of the ellipse $\frac{x^2}{36} + \frac{y^2}{16} = 1$ . If the eccentricity of the hyperbola is 5, then the length of its latus rectum is :

(A) $\frac{96}{\sqrt{5}}$

(B) $24\sqrt{5}$

(D) 16

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Let $O$ be the vertex of the parabola $x^2 = 4y$ and $Q$ be any point on it. Let the locus of the point $P$ , which divides the line segment $OQ$ internally in the ratio $2 : 3$ be the conic $C$ . Then the equation of the chord of $C$ , which is bisected at the point $(1, 2)$ , is :

(A) $5x - 4y + 3 = 0$

(B) $x - 2y + 3 = 0$

(D) $5x - y - 3 = 0$

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If the line $\alpha x + 4y = \sqrt{7}$ , where $\alpha \in \mathbb{R}$ , touches the ellipse $3x^2 + 4y^2 = 1$ at the point $P$ in the first quadrant, then one of the focal distances of $P$ is :

(A) $\frac{1}{\sqrt{3}} - \frac{1}{2\sqrt{11}}$

(B) $\frac{1}{\sqrt{3}} - \frac{1}{2\sqrt{5}}$

(D) $\frac{1}{\sqrt{3}} + \frac{1}{2\sqrt{7}}$

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Let $PQ$ be a chord of the hyperbola $\frac{x^2}{4} - \frac{y^2}{b^2} = 1$ , perpendicular to the $x$ -axis such that $OPQ$ is an equilateral triangle, $O$ being the centre of the hyperbola. If the eccentricity of the hyperbola is $\sqrt{3}$ , then the area of the triangle $OPQ$ is

(A) $2\sqrt{3}$

(B) $\frac{11}{5}$

(D) $\frac{9}{5}$

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(A) 7

(B) 9

(D) 5

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Hyperbola-JEE-Mains-PYQ’s

Mathematics Questions

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