Question

The line $lx+my+n=0$ will be a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, if

• A. $a^2{l}^2-b^2{m}^2=n^2$
• B. $a^2{l}^2+b^2{m}^2=n^2$
• C. $a{m}^2-b^2{n}^2=a^2{l}^2$
• D. None of these

Answer 1

The correct option is A $a^2{l}^2-b^2{m}^2=n^2$

If $y=Mx+C$ is tangent to hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then $C^2=a^2{M}^2-b^2$.

$lx+my+n=0$ => $y=-\frac{l}{m}x-\frac{n}{m}$

Comparing this equation with $y=Mx+C$, we get

$M=-\frac{l}{m},C=-\frac{n}{m}$

Now, $C^2=a^2{M}^2-b^2$

=> $\frac{n^2}{m^2}=a^2\frac{l^2}{m^2}-b^2$

=> $n^2=a^2{l}^2-b^2{m}^2$

Option (A)