SATHEE: Hyperbola 2 Question 7

Hyperbola 2 Question 7

7. Let $P (6, 3)$ be a point on the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ . If the normal at the point $P$ intersects the $X$ -axis at $(9, 0)$ , then the eccentricity of the hyperbola is

(2011)

(a) $\sqrt{\frac{5}{2}}$

(b) $\sqrt{\frac{3}{2}}$

(d) $\sqrt{3}$

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Answer:

Correct Answer: 7. (d)

Solution:

Equation of normal to hyperbola at $(x_{1}, y_{1})$ is

$\begin{aligned} \frac{a^{2} x}{x_{1}} + \frac{b^{2} y}{y_{1}} = (a^{2} + b^{2}) \\ ∴ At (6, 3) = \frac{a^{2} x}{6} + \frac{b^{2} y}{3} = (a^{2} + b^{2}) \\ ∵ It passes through (9, 0) . \Rightarrow \frac{a^{2} \cdot 9}{6} = a^{2} + b^{2} \\ \Rightarrow \frac{3 a^{2}}{2} - a^{2} = b^{2} \Rightarrow \frac{a^{2}}{b^{2}} = 2 \\ ∴ e^{2} = 1 + \frac{b^{2}}{a^{2}} = 1 + \frac{1}{2} \Rightarrow e = \sqrt{\frac{3}{2}} \end{aligned}$

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Hyperbola 2 Question 7

7. Let P(6,3) be a point on the hyperbola x2a2−y2b2=1. If the normal at the point P intersects the X-axis at (9,0), then the eccentricity of the hyperbola is

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7. Let $P (6, 3)$ be a point on the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ . If the normal at the point $P$ intersects the $X$ -axis at $(9, 0)$ , then the eccentricity of the hyperbola is