Answer:
a. sqrt(x^2+y^2) * (x^2+y^2) = 4xy
b. (x^2+y^2)^2 = 4x{ (x^2 - 3 y^2}
Explanation:
a) r = 2sin(2Θ)
r* = 2 sin 2 theta
We know sin 2theta = 2 sin theta cos theta
r = 2 * 2 sin theta cos theta
sin theta = y/r and cos theta = x/r
r = 4 y/r * x/r
r = 4xy / r^2
Multiply each side by r^2
r^3 = 4xy
r * r^2 = 4xy
We know r = sqrt(x^2+y^2) and r^2 = (x^2+y^2)
sqrt(x^2+y^2) * (x^2+y^2) = 4xy
b) r = 4cos(3Θ)
we know that cos 3 theta = cos^3(theta) - 3 sin^2(theta) cos(theta)
r = 4 *cos^3(theta) - 3 sin^2(theta) cos(theta)
Factor out a cos theta
r = 4 *cos(theta){ cos^2 (theta) - 3 sin^2(theta) }
We know that sin theta = y/r and cos theta = x/r
r = 4 (x/r) { (x/r)^2 - 3 (y/r)^2}
r = 4 *(x/r) { (x^2/r^2 - 3 y^2/r^2}
Multiply by r
r^2 = 4x { (x^2/r^2 - 3 y^2/r^2}
Multiply by r^2
r^2 *r^2 = 4x { (x^2 - 3 y^2}
r^4 = 4x{ (x^2 - 3 y^2}
We know r^2 = (x^2+y^2)
(x^2+y^2)^2 = 4x{ (x^2 - 3 y^2}