Mathematics - Point Question with Solution | TestHub

Given $x=a\left(\frac{1-t^2}{1+t^2}\right)$ and $y=\frac{2at}{1+t^2}$ $\begin{array}{rl}& \text{ Let }t=\tan \theta \\ & \Rightarrow x=a\left(\frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }\right)\\ & \text{ and }y=\frac{2a\tan \theta }{1+{\tan }^2\theta }\\ & \Rightarrow x=a\cos 2\theta \text{ and }y=a\sin 2\theta \\ & \Rightarrow \cos 2\theta =\frac{x}{a}\text{ and }\sin 2\theta =\frac{y}{a}\end{array}$ Squaring both sides, we get ${\cos }^{2}2\theta =\frac{x^2}{a^2}$....(i) and ${\sin }^{2}2\theta =\frac{y^2}{a^2}$....(ii) Adding Eqs. (i) and (ii), we get $\begin{array}{rl}{\cos }^{2}2\theta +{\sin }^{2}2\theta & =\frac{x^2}{a^2}+\frac{y^2}{a^2}\\ \Rightarrow x^2+y^2& =a^2\end{array}$ $\therefore$ Locus is a circle having centre at origin and radius $a$.