Question

Through the positive vertex of the hyperbola a tangent is drawn; where does it meet the conjugate hyperbola?

Answer 1

Consider the hyperbola,

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

Now, equation of tangent is

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

Given that this tangent passes through positive vertex $(a,0)$

$\therefore \frac{ax}{a^2}+\frac{0y}{b^2}=1$

$\Rightarrow x=a$

Conjugate hyperbola is given by

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

Intersecting point of tangent and the C.H is

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

$y=\pm b\sqrt{2}$

$\therefore$ Point of intersection is $(a,\pm b\sqrt{2})$.