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Find the equation of chord of contact for hyperbola $\large\frac{x^2}{a^2}-\frac{y^2}{b^2}$$=1$?

$\begin{array}{1 1}(a)\;\large\frac{xx_1}{a^2}-\frac{yy_1}{b^2}\normalsize =1\\(b)\;\large\frac{xx_1}{a^2}+\frac{yy_1}{b^2}\normalsize =1\\(c)\;\large\frac{xx_1}{b^2}-\frac{yy_1}{a^2}\normalsize =1\\(d)\;\text{None of these}\end{array}$

1 Answer

If the tangent from a point $P(x_1,y_1)$ to hyperbola $\large\frac{x^2}{a^2}-\frac{y^2}{b^2}$$=1$ touch the hyperbola at $Q(x',y')$ and $R(x",y")$ then
Equation of tangent PQ and PR are
$\large\frac{xx'}{a^2}-\frac{yy'}{b^2}$$=1$-----(1)
$\large\frac{xx"}{a^2}-\frac{yy"}{b^2}$$=1$-----(2)
Since (1) & (2) pass through $P(x_1,y_1)$ then
$\large\frac{x'x_1}{a^2}-\frac{y'y_1}{b^2}$$=1$
$\large\frac{x"x_1}{a^2}-\frac{y"y_1}{b^2}$$=1$
Hence it is clear that $Q(x',y')$ and $R(x",y")$ lie on
$\large\frac{xx_1}{a^2}-\frac{yy_1}{b^2}\normalsize =1$
Which is the equation of chord of contact QR.
Hence (a) is the correct answer.
answered Feb 10, 2014 by sreemathi.v
 

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