Question

The locus of the mid points of the chords of the standard hypererbola passing through a fixed point ( $Î\pm$, $β)$, 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 hypererbola with centre $\left(\frac{Î\pm }{2},\frac{β}{2}\right)$
• D. straight line passing through $\left(\frac{Î\pm }{2},\frac{β}{2}\right)$

Answer 1

Answer: (C)

Use $T=S_1$ which gives $\frac{xh}{a^2}−\frac{ky}{b^2}=\frac{h^2}{a^2}−\frac{k^2}{b^2}$ passes through $(Î\pm ,β)$
$⇒\frac{Î\pm h}{a^2}−\frac{kβ}{b^2}=\frac{h^2}{a^2}−\frac{k^2}{b^2}⇒\frac{{\left(x−\frac{Î\pm }{2}\right)}^2}{a^2}−\frac{{\left(y−\frac{β}{2}\right)}^2}{b^2}=\frac{1}{4}\left(\frac{{Î\pm }^2}{a^2}+\frac{{β}^2}{b^2}\right)⇒C$