Question

The locus of the middle points of the chords of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ which pass through a fixed point $(\alpha ,\beta )$ is a

• A. circle
• B. parabola
• C. hyperbola
• D. pair of straight lines

Answer 1

The correct option is C hyperbola

Let mid point of the chord is, $p(h,k)$
Thus equation of chord with P as its mid point is, $\frac{hx}{a^2}-\frac{ky}{b^2}=\frac{h^2}{a^2}-\frac{k^2}{b^2}$
Given it passes through $(\alpha ,\beta )$
$\Rightarrow \frac{h\alpha }{a^2}-\frac{k\beta }{b^2}=\frac{h^2}{a^2}-\frac{k^2}{b^2}$
$\Rightarrow \frac{{(h-\frac{\alpha }{2})}^2}{a^2}-\frac{{(k-\frac{\beta }{2})}^2}{b^2}=\frac{{\alpha }^2}{4a^2}-\frac{{\beta }^2}{4b^2}$
Hence locus of $p(h,k)$ is,
$\frac{{(x-\frac{\alpha }{2})}^2}{a^2}-\frac{{(y-\frac{\beta }{2})}^2}{b^2}=\frac{{\alpha }^2}{4a^2}-\frac{{\beta }^2}{4b^2}$