If the minimum area of the triangle formed by a tangent to the ellipse x^2b^2+y^24a^2=1 an…

Question

If the minimum area of the triangle formed by a tangent to the ellipse $\dfrac{x^2}{b^2}+\dfrac{y^2}{4a^2}=1$ and the coordinate axes is $kab$, then $k$ is equal to \_\_\_\_\_\_\_\_.

Answer: 2

Accepted Answer

Tangent at $(b\cos\theta,2a\sin\theta)$: $\dfrac{x\cos\theta}b+\dfrac{y\sin\theta}{2a}=1$; intercepts $(b\sec\theta,0)$ and $(0,2a\csc\theta)$.

$$\text{Area}=\frac12(b\sec\theta)(2a\csc\theta)=\frac{2ab}{\sin2\theta}\ \ge\ 2ab.$$

$k=2.$

Solutions · 1