Converting Polar Equations

Question

Given x^2 - y^2 = 9 (NOTE: Minus sign, not plus sign), find what r^2 equals. (Covert from Cartesian to Polar). I am pretty confused on how to solve this, so if you could go step-by-step, it would be greatly appreciated!

Answer 1

Since x = rcos(θ) and y = rsin(θ) try substituting those values into the given equation:

r²cos²(θ) - r²sin²(θ) = 9

If you factor out the r² you can solve for r.

r²(cos²(θ) - sin²(θ)) = 9

r²= 9/(cos²(θ)-sin²(θ))

At this point you can replace the denominator with cos(2θ) -- double angle trig identity.

r² = 9/cos(2θ)

To enter this equation on Desmos you have to take the square root of both sides (Desmos only allows Polar equations that are linear in "r").

Check it out here:

desmos.com/calculator/cttd012tom

Answer 2

Using x = rcos(θ) and y = rsin(θ) and substituting:

(rcos(θ))² - (rsin(θ))² = 9

r²cos²(θ) - r²sin²(θ) = 9

r²[cos²(θ) - sin²(θ)] = 9

Using the identity cos(2θ) = cos²(θ) - sin²(θ) and substituting:

r²[cos(2θ)] = 9

r² = 9/cos(2θ)

r = √[9/cos(2θ)]

r = 3/√cos(2θ)

This could be rationalized if desired:

r = 3√(cos2θ)/cos(2θ)