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  • If the chords of contact of tangents from two points <img alt="\mathrm{\left(x_{1}, y_{1}\right)}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Cleft%28x_%7B1%7D%2C%20y_%7B1%7

If the chords of contact of tangents from two points \mathrm{\left(x_{1}, y_{1}\right)} and \mathrm{\left(x_{2}, y_{2}\right)} to the hyperbola \mathrm{\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1}are at right angles, then \mathrm{\frac{x_{1} x_{2}}{y_{1} y_{2}}} is equal to

Option: 1

\mathrm{-\frac{a^{2}}{b^{2}}}


Option: 2

\mathrm{-\frac{b^{2}}{a^{2}}}


Option: 3

\mathrm{-\frac{b^{4}}{a^{4}}}


Option: 4

\mathrm{-\frac{a^{4}}{b^{4}}}


Answers (1)

best_answer

The equations of the chord of contact of tangents from \left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$ and $\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right)  to the given hyperbola are

\mathrm{\frac{x x_{1}}{a^{2}}-\frac{y y_{1}}{b^{2}}=1}\quad \ldots(1)
\mathrm{and \: \frac{\mathrm{xx}_{2}}{\mathrm{a}^{2}}-\frac{\mathrm{yy}_{2}}{\mathrm{~b}^{2}}=1}\quad \ldots(2)

lines (1) and (2) are at right angles.

\therefore \frac{b^{2} x_{1}}{a^{2} y_{1}} \times \frac{b^{2} x_{2}}{a^{2} y_{2}}=-1 \Rightarrow \frac{x_{1} x_{2}}{y_{1} y_{2}}=-\frac{a^{4}}{b^{4}}.

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Pankaj

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