Question

A tangent having slope of $-\frac{4}{3}$ to the ellipse $\frac{x^2}{18}+\frac{y^2}{32}=1$ intersects the major & minor axes in points $A$ & $B$ respectively. If $C$ is the centre of the ellipse then the area of the $△ABC$ is

• A. $12$ sq. units
• B. $24$ sq. units
• C. $36$ sq. units
• D. $48$ sq. units

Answer 1

The correct option is B $24$ sq. units

Since the major axis is along the y-axis.

$\therefore$ Equation of tangent is $x=my+\sqrt{b^2{m}^2+a^2}$

Slope of tangent $=\frac{1}{m}=\frac{-4}{3}\Rightarrow m=\frac{-3}{4}$

Hence, equation of tangent is $4x+3y=24$ or $\frac{x}{6}+\frac{y}{8}=1$

Its intercepts on the axes are $6$ and $8$.

Area $\left(\Delta AOB\right)=\frac{1}{2}\times 6\times 8=24$ sq. unit