Question

A tangent having slope $-2$ to the ellipse $\frac{x^2}{18}+\frac{y^2}{36}=1$ intersects the major axis and minor axis at $A$ and $B$, respectively. If $C(0,0)$ is the centre of the ellipse, then the area of $△ABC$ is

• A. $12$ sq. units
• B. $24$ sq. units
• C. $36$ sq. units
• D. $27$ sq. units

Answer 1

The correct option is D $27$ sq. units


Equation of tangent,
$y=mx\pm \sqrt{a^2{m}^2+b^2}$
$\Rightarrow y=-2x\pm 6\sqrt{3}$

$x=0\Rightarrow y=\pm 6\sqrt{3},A(0,\pm 6\sqrt{3})$
$y=0\Rightarrow x=\pm 3\sqrt{3},B(\pm 3\sqrt{3},0)$
$\text{area}\left(△ABC\right)=\frac{1}{2}\times AC\times BC=\frac{1}{2}\times 6\sqrt{3}\times 3\sqrt{3}=27\text{sq. units}$