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If the tangent at $θ = \frac{π}{4}$ to the curve $x = a \cos^{3} θ, y = a \sin^{3} θ$ meets the x and y axes in A and B then the area of the triangle OAB is

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a

$\frac{a^{2}}{4} sq . units$

b

$\frac{a^{2}}{2} sq . units$

c

$\frac{3 a^{2}}{4} sq . units$

d

$\frac{5 a^{2}}{4} sq . units$

answer is A.

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Detailed Solution

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$(x_{1}, y_{1}) = (a \cos^{3} θ, a \sin^{3} θ), θ = \frac{π}{4} (x_{1}, y_{1}) = (\frac{a}{2 \sqrt{2}}, \frac{a}{2 \sqrt{2}}), m = {(\frac{dy}{dx})}_{θ = \frac{π}{4}} = - 1 \Rightarrow x + y = \frac{a}{\sqrt{2}} \Rightarrow Area = \frac{c^{2}}{2 |ab|} = \frac{a^{2}}{4}$

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