Question

The letters x and y represent rectangular coordinates. Write each equation using polar coordinates $\left(r,\theta \right)$.

$y^2=2x$

Short answer

The equation using polar coordinates is $r^2{\sin }^2\theta -2r\cos \theta =0$.

Step-by-step solution

Step 1. Given Information and Concept used 

We are given an equation,

$y^2=2x$

If P is a point with polar coordinates $\left(r,\theta \right)$, the rectangular coordinates $\left(x,y\right)$ of P are given by,

$$\begin{gathered} x=r\cos \theta \\ y=r\sin \theta \end{gathered}$$

Step 2. Converting into polar coordinates  

Substituting the value of x and y,we get

$$\begin{gathered} (r\sin \theta {)}^2=2(r\cos \theta ) \\ {r}^2{\sin }^2\theta =2r\cos \theta \\ {r}^2{\sin }^2\theta -2r\cos \theta =0 \end{gathered}$$

The equation is$r^2{\sin }^2\theta -2r\cos \theta =0$.