Polar/rectangular equation converting need help!?

Question

Need to convert:

r=1-2cos2t

and

3x^2-y^2=x^2/(2x-3y)

Thanks!

Answer 1

1) We have r = sqrt(x² + y²) and tan(t) = y/x

cos(2t) = 2cos²t-1

1 + tan²t = 1/cos²t

then : 1 - 2cos(2t) = 1 - 4cos²t + 2
= 3 - 4cos²t
= 3 - 4/[1 + tan²t]
= 3 - 4/ [1 + y²/x²]
= 3 - 4x²/(x² + y²)
= (-x² + 3 y²) / (x² + y²)

and sqrt(x² + y²) = (-x² + 3y²) / (x² + y²)

or (x² + y²)^(3/2) = -x² + 3y²

2) We have x = r cost and y = r sint

so 3x² - y² = 3r² cos²t - r² sin²t = r²(3cos²t - sin²t)

and x²/(2x - 3y) = r²cos²t / [r(2cost - 3sint)]
= rcos²t / (2cost - 3sint)

then r²(3cos²t - sin²t) = rcos²t / (2cost - 3sint)

and r = cos²t / [(2cost - 3sint)(3cos²t - sin²t)]


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