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10-2. Parabola, Ellipse, Hyperbola
hard
If the tangent at a point on the ellipse $\frac{{{x^2}}}{{27}} + \frac{{{y^2}}}{3} = 1$ 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
A
$3\sqrt 3$
B
$\frac {9}{2}$
C
$9$
D
$\frac {9}{\sqrt 3}$
(JEE MAIN-2016)
Solution
Equation of tangent to ellipse
$\frac{x}{{\sqrt {27} }}\cos \theta + \frac{y}{{\sqrt 3 }}\sin \theta = 1$
Area bounded by line and co-ordinate axis
$\Delta = \frac{1}{2}.\frac{{\sqrt {27} }}{{\cos \,\theta }}.\frac{{\sqrt 3 }}{{\sin \theta }} = \frac{9}{{\sin 2\theta }}$
$\Delta = $ will be minimum when $\sin 2\theta = 1$
${\Delta _{\min }} = 9$
Standard 11
Mathematics
