Question

A tangent having slope of $-4 / 3$ to the ellipse $\frac{ x ^{2}}{18}+\frac{ y ^{2}}{32}=1$ intersects the major and minor axes at points $A$ and $B$, respectively. If $C$ is the center of the ellipse, then the area of triangle $ABC$ is

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

Answer 1

Answer: (B)

One of the tangents of slope $m$ to the given ellipse is
$y=m x+\sqrt{18 m^{2}+32}$
For $m =-4 / 3$, we have
$y=-4 / 3 x+8$
Then points on the axis where the tangents meet are $A (6,0)$ and $B (0,8)$
Then the area of triangle $ABC$ is
$\frac{1}{2}(6)(8)=24$ units