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

We have to transform the given trigonometric equation to the Cartesian form. For Cartesian coordinate we use X and Y as the axes. But for Polar coordinates we use r and $\theta$ as the axes. First, we change the polar coordinate of a point into a Cartesian coordinate. Then we put the respective Cartesian coordinate into the given equation. Then applying a formula in trigonometry related to the given equation and simplifying we will get the required equation into Cartesian form.

Formula used: $\cos 2\theta = {\cos ^2}\theta - {\sin ^2}\theta$

Complete step-by-step answer:
It is given that; polar equation is ${r^2} = {a^2}\cos 2\theta$.
We have to change the given equation into Cartesian form.
Let us consider, the coordinate of a point P on a Cartesian plane is $(x,y)$. The coordinate of the same point on the Polar plane is $(r,\theta )$.
So, the relation between the Cartesian and Polar coordinate of the same point is
$x = r\cos \theta ,y = r\sin \theta$.
So, ${r^2} = {x^2} + {y^2}$
So, we have, $x = r\cos \theta ,y = r\sin \theta$
It is given, ${r^2} = {a^2}\cos 2\theta$
We know that, $\cos 2\theta = {\cos ^2}\theta - {\sin ^2}\theta$
So, we get, ${r^2} = {a^2}({\cos ^2}\theta - {\sin ^2}\theta )$
Substitute the values we get,
\Rightarrow$$${r^2} = {a^2}(\dfrac{{{x^2}}}{{{r^2}}} - \dfrac{{{y^2}}}{{{r^2}}})$$ Simplifying we get, \Rightarrow{({r^2})^2} = {a^2}({x^2} - {y^2})$$ Substitute the values we get, $\Rightarrow{({x^2} + {y^2})^2} = {a^2}({x^2} - {y^2})$Hence, the Cartesian form is${({x^2} + {y^2})^2} = {a^2}({x^2} - {y^2})$$.

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

Note: We have to mind that, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. Let us consider, the coordinate of a point P on a Cartesian plane is $(x,y)$. The coordinate of the same point on the Polar plane is $(r,\theta )$. So, the relation between the Cartesian and Polar coordinate of the same point is,
$x = r\cos \theta ,y = r\sin \theta$
Also, we have, ${r^2} = {x^2} + {y^2}$ and $\theta = {\tan ^{ - 1}}\dfrac{y}{x}$.