Question

Transform the equation $r^2=a^2\cos 2\theta$ into Cartesian form.

• A. $x^2+y^2=a^2(x^2-y^2)$
• B. $(x^2+y^2{)}^2=a^2(x^2-y^2)$
• C. $x^2+y^2=a^2{x}^2+a^2{y}^2$
• D. None of these

Answer 1

The correct option is B $(x^2+y^2{)}^2=a^2(x^2-y^2)$

We have $x=r\cos \theta ,y=r\sin \theta$
So, $r^2=x^2+y^2$
Given $r^2=a^2\cos 2\theta$
$\Rightarrow r^2=a^2({\cos }^2\theta -{\sin }^2\theta )$
$\Rightarrow r^2=a^2(\frac{x^2}{r^2}-\frac{y^2}{r^2})$
$\Rightarrow {\left(r^2\right)}^2=a^2(x^2-y^2)$
$\Rightarrow (x^2+y^2{)}^2=a^2(x^2-y^2)$