Question

The locus of the mid points of the chords passing through a fixed point $(Î\pm ,β)$ of the hyperbola, $\frac{x^2}{a^2}−\frac{y^2}{b^2}=1$ is -

• A. a circle with centre $\left(\frac{Î\pm }{2},\frac{β}{2}\right)$
• B. an ellipse with centre $\left(\frac{Î\pm }{2},\frac{β}{2}\right)$
• C. a hyperbola with centre $\left(\frac{Î\pm }{2},\frac{β}{2}\right)$
• D. straight line through $\left(\frac{Î\pm }{2},\frac{β}{2}\right)$

Answer 1

Answer: (C)

Let mid-point of chord is $(h,k)$.
Equation of chord is $T=0$
$⇒\frac{hx}{a^2}−\frac{ky}{b^2}=\frac{h^2}{a^2}−\frac{k^2}{b^2}$
Locus is $\frac{x^2}{a^2}−\frac{y^2}{b^2}=\frac{Î\pm x}{a^2}−\frac{yβ}{b^2}$
$⇒\frac{x(x−Î\pm )}{a^2}−\frac{y(y−β)}{b^2}=0$
$⇒\frac{{\left(x−\frac{Î\pm }{2}\right)}^2−\frac{{Î\pm }^2}{4}}{a^2}−\frac{{\left(y−\frac{β}{2}\right)}^2−\frac{{β}^2}{4}}{b^2}=0$
$⇒\frac{(x−Î\pm /2{)}^2}{a^2}−\frac{(y−β/2{)}^2}{b^2}=\frac{{Î\pm }^2}{4a^2}−\frac{{β}^2}{4b^2}$