The coordinates of the point 
lying in the first quadrant on the ellipse 
so that the area of the triangle formed by the tangent at P and the coordinate axes is the smallest, are given by
Text Solution
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Any point on the ellipse is given by 
Now 
⇒ ⇒ 
⇒ ⇒ 
Hence the equation of the tangent at 
is 
Therefore, the tangent cuts the coordinates axes at the points 
and 
Thus the area of the triangle formed by this tangent and the coordinate axes is
A = 
.
But cosec2 θ θ is smallest when θ θ = 
. Therefore A is smallest when θ θ = 
.
Hence the required point is
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