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

Hint: 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 \[\theta = {\tan ^{ - 1}}\dfrac{y}{x}\].