Question

If the coordinates of a variable point $P$ be $\left(a\cos \left(\theta \right),b\sin \left(\theta \right)\right)$ where $\theta$ is a variable quantity , find the locus of $P$

Answer 1

Find the locus based on given information

Given,

Coordinates of a variable point $P$ are $\left(a\cos \left(\theta \right),b\sin \left(\theta \right)\right)$

From this we get the $x$ coordinate of the point to be $a\cos \left(\theta \right)$

From which we get

$$\begin{gathered} x=a\cos \left(\theta \right) \\ \Rightarrow \frac{x}{a}=\cos \left(\theta \right)........\left(1\right) \end{gathered}$$

Similarly for $y$ coordinate of the point to be $b\sin \left(\theta \right)$

From which we get

$$\begin{gathered} y=b\sin \left(\theta \right) \\ \Rightarrow \frac{y}{b}=b\sin \left(\theta \right)......\left(2\right) \end{gathered}$$

Squaring and adding both side of $\left(1\right)$ and $\left(2\right)$ we get

$$\begin{gathered} {\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2={\left(\cos \left(\theta \right)\right)}^2+{\left(\sin \left(\theta \right)\right)}^2 \\ \Rightarrow {\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1 \end{gathered}$$

Hence the locus of the point $P$ is ${\left(\frac{x}{a}\right)}^2+{\left(\frac{y}{b}\right)}^2=1$