The equation of the tangent to the curve $x=\operatorname{acos}^3 heta, y=\operatorname{asin}^3 heta$ at $ heta=\frac{\pi}{4}$ is

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The equation of the tangent to the curve $x=\operatorname{acos}^3 	heta, y=\operatorname{asin}^3 	heta$ at $	heta=\frac{\pi}{4}$ is

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$\begin{array}{ll} & x=\operatorname{acos}^3 	heta 	ext { and } y=\operatorname{asin}^3 	heta \ 	herefore \quad & \frac{\mathrm{~d} x}{\mathrm{~d} 	heta}=-3 \operatorname{acos}^2 	heta \sin 	heta 	ext { and } \frac{\mathrm{d} y}{\mathrm{~d} 	heta}=3 \operatorname{asin}^2 	heta \cos 	heta \ 	herefore \quad & \frac{\mathrm{~d} y}{\mathrm{~d} x}=\frac{\frac{\mathrm{d} y}{\mathrm{~d} 	heta}}{\frac{\mathrm{~d} x}{\mathrm{~d} 	heta}}=-	an 	heta \ 	herefore \quad & \left(\frac{\mathrm{d} y}{\mathrm{~d} x}ight)_{\left(	heta=\frac{\pi}{4}ight)}=-1 \ & 	ext { At } 	heta=\frac{\pi}{4} \ & x=\mathrm{a} \cos ^3 \frac{\pi}{4}=\frac{\mathrm{a}}{2 \sqrt{2}} \ y & =a \sin ^3 \frac{\pi}{4}=\frac{\mathrm{a}}{2 \sqrt{2}}\end{array}$ $	herefore \quad$ Equation of the tangent at $\left(\frac{\mathrm{a}}{2 \sqrt{2}}, \frac{\mathrm{a}}{2 \sqrt{2}}ight)$ is $\begin{aligned} & y-\frac{a}{2 \sqrt{2}}=-1\left(x-\frac{a}{2 \sqrt{2}}ight) \ & \Rightarrow x+y=\frac{a}{\sqrt{2}} \end{aligned}$

The equation of the tangent to the curve $x=\operatorname{acos}^3 \theta, y=\operatorname{asin}^3 \theta$ at…

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