If the tangent at a point on the ellipse `x^2/27+y^2/3=1` meets the coordinate axes at `A` and `B,` and the origin, then the minimum area (in sq. units) of the triangle `OAB` is:A. 9B. `(9)/(2)`C. `9sqrt(3)`D. `3sqrt(3)`

If the tangent at a point on the ellipse `x^2/27+y^2/3=1` meets the co

1.	If the tangent at a point on the ellipse `x^2/27+y^2/3=1` meets the coordinate axes at `A` and `B,` and the origin, then the minimum area (in sq. units) of the triangle `OAB` is:A. 9B. `(9)/(2)`C. `9sqrt(3)`D. `3sqrt(3)`
Answer» Correct Answer - A Let `P(3sqrt(3) cos theta, sqrt(3) sin theta)` be a point on the ellipse. Then, the equation of the tangent at P is `(x)/(3sqrt(3)) cos theta +(y)/(sqrt(3)) sin theta =1`. This meets the coordinate axes at `A(3sqrt(3) sec theta, 0) and B(0, sqrt(3) " cosec" theta)`. Let `Delta` be the area of `Delta OAB.` Then, `Delta=(1)/(2)(OAxxOB)=(1)/(2) xx (3sqrt(3))/(cos theta)xx (sqrt(3))/(sin theta)=(9)/(sin2theta)` Clearly, `Delta` is minimum when `sin 2 theta` is maximum. The maximum value of `sin 2 theta` is 1. Hence, the minimum value of `Delta` is 9.

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