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The minimum area of triangle formed by t...

The minimum area of triangle formed by the tangent to the ellipse `x^(2)/a^(2)+y^(2)/b^(2)=1` and coordinates axes is

A

ab sq. units

B

`(a^(2)+b^(2))/(2)` sq. units

C

`((a+b)^(2))/(2)` sq. units

D

`(a^(2)+ab+b^(2))/(3)` sq. units

Text Solution

Verified by Experts

Tangent to the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
at `p( a cos theta, b sin theta)` is given by
`(xcos theta)/(a)+(y sin theta)/(b)=1`
It meets he coordinate axes at `A(a sec theta, 0) and B(0, b "coses" beta)`
or `Delta=(ab)/(sin 2 theta)`
For area to be minimum, `sin theta` should be maxiumum and we know that teh maxiumum value of `sin theta` is 1. Therefore,
`Delta_("max")=ab`
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