Question

Let \(P(4,3)\) be a point on the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 .\) If the normal at \(P\) intersects the \(X\)-axis at \((16,0)\), then the eccentricity of the hyperbola is
(A) \(\frac{\sqrt{5}}{2}\)
(B) 2
(C) \(\sqrt{2}\)
(D) \(\sqrt{3}\)

Answer 1

Ans: (B)
Hint: Normal at \(P(4,3)\)
$$
\begin{aligned}
&\frac{a^{2} x}{4}+\frac{b^{2} y}{3}=a^{2}+b^{2} \text { through }(16,0) \\
&\Rightarrow 4 a^{2}=a^{2}+b^{2} \\
&\Rightarrow \frac{b^{2}}{a^{2}}=3 \therefore e=2
\end{aligned}
$$

Answer 2

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