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=0⇒dydx=−9x4y
⇒dydx∣∣∣(√8cosθ,√18sinθ)=−9√8cosθ4√18sinθ
=−√92cotθ
Hence, the equation of the tangent at (√8cosθ,√18sinθ) is
y−√18sinθ=−√92cotθ(x−√8cosθ)
Thus, the area of the triangle formed by this tangent and the coordinate axes is
A=12√18⋅√8⋅1cosθsinθ=6cosθsinθ=12cosec2θ
But cosec2θ is smallest when θ=π4,therefore A is smallest when θ=π/4.