Question

Convert (xy)^2=9 to an equation in polar coordinates [answer]=r^18

Answer 1

In polar coordinates,
`r^18 = 2^9.`

Polar coordinates: We convert x and y to r and t using x = r cos(t) and y = r sin(t).

Substitute: Replace xy with
`r^2 sin(t) cos(t`) in the given equation.

Simplify: Raise both sides to the power of 9 and expand using DeMoivre's Theorem:
`(r^2 sin(t) cos(t))^9 = 2^9.` This simplifies to
`r^18 sin(18t) cos(18t) = 2^9.`

Magnitude: Since the magnitude (r) doesn't depend on the angle (t), we can isolate it by taking the absolute value of both sides. This leads to
`|r^18| = 2^9.`

Therefore,
`r^18 = 2^9`