Question

The number of possible tangent(s) drawn to the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$, which is/are perpendicular to $5x+2y=10$, is

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

The correct option is A $0$

Given equation of the hyperbola is,
$\frac{x^2}{9}-\frac{y^2}{4}=1$
Differentiating w.r.t. $x$, we get
$\frac{2x}{9}-\frac{2y{y}^{\prime }}{4}=0$
Slope of the tangent at $(3\sec \theta ,2\tan \theta )$ is,
$y^{\prime }=\frac{12\sec \theta }{18\tan \theta }=\frac{2\sec \theta }{3\tan \theta }=\frac{2}{3\sin \theta }$

Slope of the line $5x+2y=10$ is, $m=-\frac{5}{2}$
$\therefore$ Slope of any line perpendicular to $5x+2y=10$ is, $m_1=\frac{2}{5}$
So, $\frac{2}{5}=\frac{2}{3\sin \theta }$
$\Rightarrow \sin \theta =\frac{5}{3}$, which is not possible.
Hence, no such tangent is possible.