15:I[8216,["972","static/chunks/972-b28da0de92681928.js","47","static/chunks/47-a7f06e9eda1ac756.js","115","static/chunks/app/%5Bexam%5D/questions/%5Byear%5D/%5Bsubject%5D/%5Bchapter%5D/%5BquestionId%5D/page-7b6f0a12b9cc0206.js"],"BackButton"] 16:I[1553,["972","static/chunks/972-b28da0de92681928.js","47","static/chunks/47-a7f06e9eda1ac756.js","115","static/chunks/app/%5Bexam%5D/questions/%5Byear%5D/%5Bsubject%5D/%5Bchapter%5D/%5BquestionId%5D/page-7b6f0a12b9cc0206.js"],"QuestionViewer"] 17:T4f4,

We have, $$x^{2 / 3}+y^{2 / 3}=a^{2 / 3}$$

$$\begin{aligned} \text{Let} \quad & x=a \cos ^3 \theta \text { and } y=a \sin ^3 \theta \\ & \frac{d x}{d \theta}=-3 a \cos ^2 \theta \sin \theta \end{aligned}$$

and $$\quad \frac{d y}{d \theta}=3 a \sin ^2 \theta \cos \theta$$

$$\begin{aligned} \frac{d y}{d x} & =\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}=\frac{3 a \sin ^2 \theta \cos \theta}{-3 a \cos ^2 \theta \sin \theta} \\ & =-\tan \theta \end{aligned}$$

Equation of tangent is

$$\begin{aligned} & \left(y-a \sin ^3 \theta\right)=-\tan \theta\left(x-a \cos ^3 \theta\right) \\ & y \cos \theta-a \sin ^3 \theta \cos \theta=-x \sin \theta+a \cos ^3 \theta \sin \theta \\ & x \sin \theta+y \cos \theta=a \sin \theta \cos \theta\left(\cos ^2 \theta+\sin ^2 \theta\right) \\ & \frac{x}{a \cos \theta}+\frac{y}{a \sin \theta}=1 \end{aligned}$$

$$\Rightarrow a \cos \theta$$ is $$x$$-intercept and $$a \sin \theta$$ is $$y$$-intercept

Let $$A(a \cos \theta, 0)$$ and $$B$$ is $$(0, a \sin \theta)$$ are the points, where the tangents cuts $$X$$-axis and $$Y$$-axis, respectively.

$$A B=\sqrt{a^2 \cos ^2 \theta+a^2 \sin ^2 \theta}=a$$

$$\therefore$$ The required position of tangent is constant.

Application Of Derivatives

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