Question: Find the polar equation for the ellipse with one focus at the pole.
A ) r = 125 + 3 cos θ
B ) r = 65 + 3 cos θ
C ) r = 33 + 5 cos θ
D ) None of the above
Steps to derive
1 P′ (3, π) and P (0.75, 0) are the vertices of the ellipse. Let C be the center of the ellipse.
2 Length of the major axis, PP′ is 2a = (154) and hence semimajor axis a = 158.
3 a - c = 0.75
[The vertex (0.75,0) is 0.75 units right of the pole (Focus).]
4 158 - c = 0.75
5 c = 98
6 Eccentricity of the ellipse is e = ca = (98)(158) = 35
7 k = (a - c)(1+ee) = 0.75 × 83 = 2
[Substitute the values of e, a & c.]
8 The polar equation of the ellipse is r = (2)(35)1+35cos θ = 65+3 cos θ.
[The polar equation of the conic is r = ke1+e cos θ.]
Hence the right answer is Option B
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