Question

The point of intersections of tangents drawn to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at the points where it intersected by the line $lx+my+n=0$, is

• A. $(\frac{-a^{2}l}{n},\frac{b^{2}m}{n})$
• B. $(\frac{-a^{2}l}{m},\frac{b^{2}n}{m})$
• C. $(\frac{a^{2}l}{m},\frac{-b^{2}n}{m})$
• D. $(\frac{a^{2}l}{m},\frac{b^{2}n}{m})$

Answer 1

The correct option is A $(\frac{-a^{2}l}{n},\frac{b^{2}m}{n})$

REF.Image

Given

Hyperbola is given by

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

AB as a chord of contact

$T=0$

$\frac{x\alpha }{a^2}-\frac{y\beta }{b^2}=1$ ___(i)

But given $E{q}^n$ of AB is $lx+my+n=0$ ___ (ii)

$\frac{\alpha }{a^{2}l}=\frac{-\beta }{b^{2}m}=\frac{1}{-n}$

$\Rightarrow \alpha =\frac{-a^{2}l}{n},\beta =\frac{b^{2}m}{n}$

$P(\frac{-a^{2}l}{n},\frac{b^{2}m}{n})$