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Question

The coordinates of the point P(x,y) lying in the first quadrant on the ellipse x28+y218=1 so that the area of the triangle formed by the tangent at P and the coordinate axes is the smallest, are given by

A
(2,3)
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B
(8,0)
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C
(0,18)
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D
none of these
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Solution

The correct option is A (2,3)
Any point on the ellipse x28+y218=1 is given by (8cosθ,18sinθ)

Now, 2x8+218ydydx=0dydx=9x4y

dydx(8cosθ,18sinθ)=98cosθ418sinθ

=92cotθ

Hence, the equation of the tangent at (8cosθ,18sinθ) is

y18sinθ=92cotθ(x8cosθ)

Thus, the area of the triangle formed by this tangent and the coordinate axes is

A=121881cosθsinθ=6cosθsinθ=12cosec2θ

But cosec2θ is smallest when θ=π4,therefore A is smallest when θ=π/4.
Hence the required point is
(812,1812)(2,3)

Ans: A

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