If the tangent at a point on the ellipse x^227+y^23=1 meets the coordinate axes at A and B…

Question

If the tangent at a point on the ellipse $\dfrac{x^2}{27}+\dfrac{y^2}3=1$ meets the coordinate axes at $A$ and $B$, and $O$ is the origin, then the minimum area (in sq. units) of the triangle $OAB$ is:

Accepted Answer

Equation of tangent to ellipse

$${x \over {\sqrt {27} }}$$ cos$$\theta $$ + $${y \over {\sqrt 3 }}$$sin$$\theta $$ = 1

Area bounded by line and co-ordinate axis

$$\Delta $$ = $${1 \over 2}$$ . $${{\sqrt {27} } \over {\cos \theta }}.{{\sqrt 3 } \over {\sin \theta }}$$ = $${9 \over {\sin 2\theta }}$$

$$\Delta $$ = will be minimum when sin 2$$\theta $$ = 1

$$\Delta $$min = 9

Solutions · 1