Question

The total number of tangents to the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ that are perpendicular to the line $5x+2y-3=0$ is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is A $0$

Equation of tangnet $y=mx\pm \sqrt{a^2{m}^2-b^2}$
So $a^2{m}^2-b^2>0\Rightarrow 9m^2-4\ge 0$
$\Rightarrow (3m-2)(3m+2)\ge 0$
$\therefore m\in (-\infty ,\frac{-2}{3}]\cup [\frac{2}{3},\infty )$
Slope of given line is $\frac{-5}{2}$
$\therefore$ Slope of perpendicular tangnets is $\frac{2}{5}$
Hence no perpedicular tangnet exists.

Alternate Solution:
Equation of the tangent to the hyperbola in parametric form is
$\frac{x\sec \theta }{3}-\frac{y\tan \theta }{2}=1$
$\because$ tangent is perpendicular to the line $5x+2y=3$
$\therefore \frac{2\sec \theta }{3\tan \theta }\times \frac{-5}{2}=-1\Rightarrow \sin \theta =\frac{5}{3}$
Hence no tangent possible