Q. Tangent to the ellipse $\frac{x^{2}}{32} + \frac{y^{2}}{18} = 1$ having slope $- \frac{3}{4}$ meet the coordinate axes in $A$ and $B$. Find the area of the $\mathrm{\Delta}AOB,$ where $O$ is the origin
A) $12\$sq unit
B) 8 sq unit
C) $24\$sq unit
D) 32 sq unit
Explanation
(c)
Equation of tangent with slope $- \frac{3}{4}$ is
$y = - \frac{3}{4}x + c$
According to condition of tangency
$c = \sqrt{32 \times \left( \frac{- 3}{4} \right)^{2} + 18}$
$= \sqrt{18 + 18} = 6$
$\therefore\ y = - \frac{3}{4}x + 6$
$\Rightarrow \ \ 4y + 3x = 24\$
It meets the coordinate axes in $A$ and $B$
$\therefore A \equiv (8,0)$ and$\ B \equiv (0,6)$
Required area $= \frac{1}{2} \times 8 \times 6 = 24$ sq unit
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.