Given: (x - 1)^2 - (y + 5)^2 = -24(x−1)2−(y+5)2=−24
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Convert to polar coordinates.
Expand the squares:
x^2 -2x + 1 - (y^2 + 10y + 25) = -24x2−2x+1−(y2+10y+25)=−24
Regroup by power:
x^2 - y^2 -2x - 10y + 1 - 25 = -24x2−y2−2x−10y+1−25=−24
Combine the constant terms:
x^2 - y^2 -2x - 10y = 0x2−y2−2x−10y=0
Substitute rcos(theta)rcos(θ) for x and rsin(theta)rsin(θ) for y:
(rcos(theta))^2 - (rsin(theta))^2 -2(rcos(theta)) - 10(rsin(theta)) = 0(rcos(θ))2−(rsin(θ))2−2(rcos(θ))−10(rsin(θ))=0
Lets move the factors of r outside the ():
(cos^2(theta) - sin^2(theta))r^2 -(2cos(theta) + 10sin(theta))r = 0(cos2(θ)−sin2(θ))r2−(2cos(θ)+10sin(θ))r=0
There are two roots, r = 0r=0 which is trivial should be discarded, and:
(cos^2(theta) - sin^2(theta))r -(2cos(theta) + 10sin(theta)) = 0(cos2(θ)−sin2(θ))r−(2cos(θ)+10sin(θ))=0
Solve for r:
r = (2cos(theta) + 10sin(theta))/(cos^2(theta) - sin^2(theta))r=2cos(θ)+10sin(θ)cos2(θ)−sin2(θ)
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