Question

The condition that the line $lx+my+n=0$ may be a tangent to the rectangular hyperbola $xy=c^2$ is

• A. $a^2{l}^2+b^2{m}^2=n^2$
• B. $a{m}^2+ln$
• C. $a^2{l}^2-b^2{m}^2=n^2$
• D. $4c^{2}lm=n^2$

Answer 1

The correct option is D $4c^{2}lm=n^2$

The line $lx+my+n=0$ has to be the tangent to $xy=c^2$

Substituting $y=\frac{-lx-n}{m}$ in the equation, we have

$x\left(\frac{-lx-n}{m}\right)=c^2$

$\therefore -l{x}^2-nx-m{c}^2=0$

$\therefore l{x}^2+nx+m{c}^2=0$

Since the line is a tangent, there has to be only one value of $x$.

So, the dicriminant has to be zero.

$\therefore n^2-4l\times m{c}^2=0$

$\therefore n^2=4c^{2}lm$