HyperbolaHardBloom L3
Question
Locus of the point of intersection of the tangents at the points with eccentric angles $\phi$ and $\frac{\pi}{2} - \phi$ on the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ is :
Options
A.$x = a$
B.$y = b$
C.$x = b$
D.$y = ab$
Solution
Sol. (b) Tangent at $\phi$ is
$$\begin{array}{r} \frac{xsec\phi}{a} - \frac{ytan\phi}{b} = 1\#(1) \end{array}$$
Tangent at $\frac{\pi}{2} - \phi$ is
$$\begin{array}{r} \frac{xcosec\phi}{a} - \frac{ycot\phi}{b} = 1\#(2) \end{array}$$
(1) $\times cosec\phi - (2) \times sec\phi$
$$\begin{matrix} & \ \Rightarrow \ \frac{y}{b}\left( \frac{1}{sin\phi} - \frac{1}{cos\phi} \right) = \frac{1}{sin\phi} - \frac{1}{cos\phi} \\ & \ \Rightarrow y = b \end{matrix}$$
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