Find an equation equivalent to x^2 - y^2 = 4 in polar coordinates.

Question

Find an equation equivalent to x^2 - y^2 = 4 in polar coordinates.

 

Answer 1

x^2-y^2=4

x=rcosθ

y=rsinθ

substitute

r^2cos^2θ-r^2sin^2θ=4

sin^2θ+cos^2θ=1

therefore cos^2θ=1-sin^2θ

substitute

r^2(1-sin^2θ)-r^2sin^2θ=4

r^2-r^2sin^2θ-r^2sin^2θ=4

r^2-2r^2sin^2θ=4

r^2(1-2sin^2θ)=4

r^2=4/(1-2sin^2θ)

take the square root of both sides and separate the numerator and denominator in the square root

r=√4/√(1-2sin^2θ)

r=2/√(1-2sin²θ)

 

Answer 2

Hi Dakota,

1. Substitute for x and y in your equation: x=r*cosθ and y=r*sinθ
2. You get r²cos²θ - r²sin²θ = 4
3. Factor out r² on the left side -> r²(cos²θ - sin²θ)=4

4. Using the double angle formula you get r²cos(2θ)=4

5. I leave it to you to solve for r.

 

Hope this helps!