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Q.
The locus of the mid points of the chords passing through a fixed point $(\alpha, \beta)$ of the hyperbola, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is -
Conic Sections
Solution:
Let mid-point of chord is $( h , k )$.
Equation of chord is $T=0 $
$\Rightarrow \frac{h x}{a^2}-\frac{k y}{b^2}=\frac{h^2}{a^2}-\frac{k^2}{b^2}$
Locus is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{\alpha x}{a^2}-\frac{y \beta}{b^2}$
$\Rightarrow \frac{x(x-\alpha)}{a^2}-\frac{y(y-\beta)}{b^2}=0$
$\Rightarrow \frac{\left(x-\frac{\alpha}{2}\right)^2-\frac{\alpha^2}{4}}{a^2}-\frac{\left(y-\frac{\beta}{2}\right)^2-\frac{\beta^2}{4}}{b^2}=0$
$\Rightarrow \frac{(x-\alpha / 2)^2}{a^2}-\frac{(y-\beta / 2)^2}{b^2}=\frac{\alpha^2}{4 a^2}-\frac{\beta^2}{4 b^2}$