Question

Rectangular to Polar Sketch a graph of the rectangular equation

${(x^2+y^2)}^3=4x^2{y}^2$

Short answer

Graph of given equation is-

Step-by-step solution

Step 1. Given information

An equation is given as ${(x^2+y^2)}^3=4x^2{y}^2$

Step 2. Concept used

First convert the equation into polar equation then plot graph accordingly. For conversion of equation, use formula-

$r=\sqrt{x^2+y^2}$

Step 3. Calculation

Equation is given as ${(x^2+y^2)}^3=4x^2{y}^2$,

Substitute the value as$x=r\cos \theta \&y=r\sin \theta$ by using the formula.$r=\sqrt{x^2+y^2}$

$\begin{array}{l}{(x^2+y^2)}^3=4x^2{y}^2\\ {\left(r^2\right)}^3\text{Invalid <m:msup> element}\text{Invalid <m:msup> element}=4\\ r^6=4\left(r^2{\cos }^2\theta \right)\left(r^2{\sin }^2\theta \right)\end{array}$

Take square root on both sides,

$\begin{array}{l}\sqrt{r^6}=\sqrt{4r^4{\cos }^2\theta {\sin }^2\theta }\\ r^3=2r^2\left(\cos \theta \sin \theta \right)\\ r=2\left(\cos \theta \sin \theta \right)\\ r=\sin 2\theta \end{array}$

Graph of above equation is-