Question

Tangents to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ make angles ${\theta }_1,{\theta }_2$ with the transverse axis. If ${\theta }_1,{\theta }_2$ are
complementary then the locus of the point of intersection of the tangents is

• A.
  
• B.
  
  
• C.
  
  
• D.
  
  

Answer 1

The correct option is A


${\theta }_1+{\theta }_2={90}^{\circ }\Rightarrow tan{\theta }_{1}tan{\theta }_2=1\Rightarrow m_1{m}_2=1\Rightarrow \frac{y_1^2+b^2}{x_1^2-a^2}=1\Rightarrow x_1^2-y_1^2=a^2+b^2$
$\therefore x^2-y^2=a^2+b^2$