Question

If the tangent at the point $(a\sec \varphi ,b\tan \varphi )$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ meets the transverse axis at $T$, then the distance of $T$ from the focus of the hyperbola is :

• A. $a(e+\cos \varphi )$
• B. $a(e-\cos \varphi )$
• C. $b(e+\cos \varphi )$
• D. None of these

Answer 1

The correct option is B $a(e-\cos \varphi )$

Equation of tangent at the point $(a\sec \varphi ,b\tan \varphi )$ is $\frac{x\sec \varphi }{a}-\frac{y\tan \varphi }{b}=1$

Finding its intersection with transverse axis $(y=0)$ we get

$\therefore$ Distance of the point $T$ from the focus $(ae,0)$

$=ae-a\cos \varphi [\because a\cos \varphi \le a,ae>a]$