Convert the polar equation to rectangular form r2 2cos 2 th

Convert the polar equation to rectangular form. r^2 = 2cos 2 theta

Solution

r^2 = 2cos2theta

= 2(cos^2theta - sin^2theta)

plug x = rcostheta ; y = rsintheta

r^2 = 2(x^2/r^2 - y^2/r^2)

r^4 = 2x^2 - 2y^2

we know r^2 = x^2 +y^2

So, (x^2 +y^2)^2 = 2x^2 - 2y^2 (rectangular coordinates)