Question

The locus of the mid-point of the chords of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passing through a fixed point $(\alpha ,\beta )$ is a hyperbola with centre at $(\frac{\alpha }{2},\frac{\beta }{2})$ :

• A. $\frac{(x-\alpha /2{)}^2}{a^2}-\frac{(y-\beta /2{)}^2}{b^2}=\frac{{\alpha }^2}{4a^2}-\frac{{\beta }^2}{4b^2}$
• B. $\frac{(x+\alpha /2{)}^2}{a^2}-\frac{(y-\beta /2{)}^2}{b^2}=\frac{{\alpha }^2}{4a^2}-\frac{{\beta }^2}{4b^2}$
• C. $\frac{(x-\alpha /2{)}^2}{a^2}-\frac{(y-\beta /2{)}^2}{b^2}=\frac{{\alpha }^2}{4a^2}+\frac{{\beta }^2}{4b^2}$
• D. None of these

Answer 1

Answer: (A)

$\frac{(x-\alpha /2{)}^2}{a^2}-\frac{(y-\beta /2{)}^2}{b^2}=\frac{{\alpha }^2}{4a^2}-\frac{{\beta }^2}{4b^2}$

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,
$\frac{{(x-\frac{\alpha }{2})}^2}{a^2}-\frac{{(y-\frac{\beta }{2})}^2}{b^2}=\frac{{\alpha }^2}{4a^2}-\frac{{\beta }^2}{4b^2}$