Q. Let $(\alpha,\ \beta)$ be a point from which two perpendicular tangents can be drawn to the ellipse $4x^{2} + 5y^{2} = 20.$ If $F = 4\alpha + 3\beta$, then
A) $- 15 \leq F \leq 15$
B) $F \geq 0$
C) $- 5 \leq F \leq 20$
D) $F \leq - 5\sqrt{5}$ or $F \geq 5\sqrt{5}$
Explanation
(a)
$(\alpha,\ \beta)$ lies on the director circle of the ellipse $ie$, on $x^{2} + y^{2} = 9$
So, we can assume
$\alpha = 3\cos\theta\ ,\beta = 3\sin\theta$
$\therefore F = 12\cos\theta + 9\sin\theta = 3\left( 4\cos\theta + 3\sin\theta \right)$
$\Rightarrow - 15 \leq F \leq 15$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.