How do you find the rectangular equation for r=2costhetar=2cosθ?

1 Answer
Douglas K.
Nov 7, 2016

Please see the explanation for steps leading to the rectangular equation.

Explanation:

Multiply both sides by r:

r^2 = 2rcos(theta)r2=2rcos(θ)

Substitute (x^2 + y^2)(x2+y2) for r^2r2 and x for rcos(theta)rcos(θ):

x^2 + y^2 = 2xx2+y2=2x

Add h^2 - 2xh2−2x to both sides:

x^2 - 2x + h^2 + y^2 = h^2x2−2x+h2+y2=h2

Use the right side of the pattern (x - h)^2 = x^2 - 2hx + h^2(x−h)2=x2−2hx+h2 equal to the first 3 terms to find the value of h:

x^2 - 2hx + h^2 = x^2 - 2x + h^2x2−2hx+h2=x2−2x+h2

-2hx = -2x−2hx=−2x

h = 1h=1

Substitute the left side of the pattern with h = 1 and the right side becomes 1^212

(x - 1)^2 + y^2 = 1^2(x−1)2+y2=12

It is a circle with center of (1, 0)(1,0) and radius of 1:

Here is a graph

enter image source here

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