Question

Find the locus the mid-point of focal chords of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1?$

• A.
• B.
• C.
• D.

Answer 1

Answer: (C), (D)

The correct options are
C

D

Let p(h,k) be the mid-point of the focal chord

Equation of chord whose mid-point is given

$T=S_1$

$\frac{xh}{a^2}-\frac{ky}{b^2}-1=\frac{h^2}{a^2}-\frac{k^2}{b^2}-1$ .........(1)

Since, this chord passes through focus, either (ae,0) or (-ae,0) when chord passes through (ae,0) it

should satisfty the equation (1)

$\frac{ae\times h}{a^2}-\frac{k\times o}{b^2}=\frac{h^2}{a^2}-k^2{b}^2$

$\frac{eh}{a}=\frac{h^2}{a^2}-\frac{k^2}{b^2}$

Then, locus is $\frac{ex}{a}=\frac{x^2}{a^2}-\frac{y^2}{b^2}$

if it passes through (-ae,0)

Substituting (-ae,0) in the equation (1)

$\frac{-aeh}{a^2}-0=\frac{h^2}{a^2}-\frac{k^2}{b^2}$

$\therefore$ locus is $\frac{-ex}{a}=\frac{x^2}{a^2}-\frac{y^2}{b^2}$