Question

A tangent having slope $-\frac{4}{3}$ to the ellipse $\frac{x^2}{18}+\frac{y^2}{32}=1$ meets the major and minor axes at $A$ and $B$. If $O$ is the centre of the ellipse then the area of $△OAB$ is:

• A. $16$ sq units
• B. $20$ sq. units
• C. $24$ sq. units
• D. $22$ sq. units

Answer 1

Answer: (C)

$24$ sq. units

A line $y=mx+c$ is tangent to ellipse only and only if $c^2=a^2{m}^2+b^2$

Using above equation we have $c^2=18\times \frac{16}{9}+32$

Which gives as $c=8$

Now equation of tangent is $y=\frac{-4}{3}x+8$

So this line will intersect major and minor axis at $(6,0)$ and $(0,8)$

So area of triangle $△OAB$ is $24.$