A point P can be located on an X-Y coordinate plane with respect to the origin and can be plotted easily and is written in the form of (x, y). This is known as the rectangular form. But sometimes it is easier to locate the same point or to graph an equation in the polar coordinate form which is the form written with respect to the radius and the angle ‘θ’, written as (r, θ). There are different types of polar graphs and they form circles, cardioids, rose curves etc.
Example 1: Convert the polar equation, r = 4 to rectangular form and name the shape of its graph.
Square the given equation on both sides. This gives: r2= 16
The important conversions from polar form to rectangular form are x = r cosθ, y = r sinθ and hence x2 + y2 = r2
Hence we get: x2+ y2= 16.
This is the equation of a circle and it is already written in its standard form -> x2+ y2= r2.
Therefore, the center of the circle is (0, 0) and its radius is 4.
Example 2: Convert the polar equation, r = 2cosθ to rectangular form and name the shape of its graph.
Multiply by ‘r’ on both sides. This gives: r2= 2rcosθ
We know that, x= r cosθ and x2+ y2 = r2
Hence we get: x2+ y2= 2x==> x2 – 2x+ y2= 0.
Using completing the squares method, we get: (x – 1)2 + y2 = 1
This is the equation of a circle, written in the form of (x- h)2 + (y- k)2= r2
Therefore, the center of the circle is (1, 0) and its radius is 1.