Question

The point of intersection of two tangents to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$ the product of whose slopes is $c^2$, lies on the curve.

• A.
  
• B.
  
• C.
  
• D.
  

Answer 1

The correct option is C


Let the slopes of the two tangents to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be cm and c/m
Then the equation of the tangents are y = $cmx+\sqrt{a^2{c}^2{m}^2-b^2}\to \left(1\right)$
and my $-cx=\sqrt{a^2{c}^2-b^2{m}^2}\to \left(2\right)$
Squaring and subtracting (2) from (1) we get $(y-cmx{)}^2-(my-cx{)}^2=a^2{c}^2{m}^2-b^2-a^2{c}^2+b^2{m}^2$
$\Rightarrow (1-m^2)(y^2-c^2{x}^2)=-(1-m^2)(a^2{c}^2+b^2)\Rightarrow y^2+b^2=c^2(x^2-a^2)$