Question

The minimum area of triangle formed by the tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and coordinate axes is:

• A. $ab$ sq. units
• B. $\frac{a^2+b^2}{2}$ sq. units
• C. $\frac{(a+b{)}^2}{2}$ sq. units
• D. $\frac{a^2+ab+b^2}{3}$sq. units

Answer 1

The correct option is A $ab$ sq. units

The equation of tangent at $(a\cos \theta ,b\sin \theta )$ is,

$\frac{x\cos \theta }{a}+\frac{y\sin \theta }{b}=1$

It meets the coordinate axes at $A(0,b\cos ec\theta ),B=(a\sec \theta ,0)$

Thus area of triangle is $=\frac{ab}{2\sin \theta \cos \theta }=\frac{ab}{\sin 2\theta }\ge ab.$