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Question: How do you convert \({x^2} + {y^2} - 2ax\) to polar form?...

How do you convert x2+y2−2ax{x^2} + {y^2} - 2ax to polar form?

Explanation

Solution

The given equation is in the Cartesian form and we have to convert it into polar form.
Polar form is another (Cartesian form is also the one) method of representing the coordinates in space. Polar form has angles to represent with the x –y plane.
Using the above definition we will convert the given Cartesian equation in polar form.

Complete step by step answer: Let’s discuss more about polar coordinates in order to solve the given problem.
Unlike Cartesian coordinates, for polar coordinates we do not have to move on a straight line of x- coordinates or y- coordinates but we use to make an angle with reference direction at reference point. The reference direction is called the polar axis and the reference point is called the pole. The distance of the polar axis from the origin is represented by ‘r’ and angle is represented by θ\theta according to the horizontal (rcos⁡θr\cos \theta ) and vertical (rsin⁡θr\sin \theta ) component of ‘r’.
Now, we will do the conversion of the Cartesian equation given to us in the question.
Let us assume;
⇒x=rcos⁡θ\Rightarrow x = r\cos \theta
⇒y=rsin⁡θ\Rightarrow y = r\sin \theta
(We assume x as horizontal component and y as vertical component)
x2+y2−2ax{x^2} + {y^2} - 2ax....................1
On substituting the values of x and y in equation 1
⇒(rcos⁡θ)2+(rsin⁡θ)2−2arcos⁡θ=0\Rightarrow {\left( {r\cos \theta } \right)^2} + {\left( {r\sin \theta } \right)^2} - 2ar\cos \theta = 0
⇒(r2cos⁡2θ)+(r2sin⁡2θ)=2arcos⁡θ\Rightarrow \left( {{r^2}{{\cos }^2}\theta } \right) + \left( {{r^2}{{\sin }^2}\theta } \right) = 2ar\cos \theta (We will cancel the common terms of the equation)
⇒r2(cos⁡2θ+sin⁡2θ)=2arcos⁡θ\Rightarrow {r^2}({\cos ^2}\theta + {\sin ^2}\theta ) = 2ar\cos \theta (sin⁡2θ+cos⁡2θ=1{\sin ^2}\theta + {\cos ^2}\theta = 1)
⇒r−2acos⁡θ=0\Rightarrow r - 2a\cos \theta = 0 (We have cancelled r from both sides)

Note:
Polar coordinate form is helpful in many mathematical applications such as solving double and triple integrals for making the calculation simpler, polar coordinates are used for navigation purposes in both sea and air, the latest and most usable application is GPS (Global positioning system) .