Question: Find the polar equation for the ellipse with focus at the pole and [3 / 2, 0] and (6, π) as the end points of its major axis.

A ) r = 123 + 5 cos θ
B ) r = 33 + 12 cos θ
C ) r = 512 + 3 cos θ
D ) r = 125 + 3 cos θ

Steps to derive

1 P (3 / 2, 0) and P′ (6, π) are the vertices of the ellipse.


2 C is the center of the ellipse.
 [Mid point of PP′.]

3 F is the focus, which is at pole.

4 Length of the major axis = PP′ = 2a = 2(15 / 4) = 15 / 2

5 So, semimajor axis is a = (15 / 4)

6 The vertex (3 / 2, 0) is 3 / 2 units right of the pole (Focus).

7 a - c = 32

8 154 - c = 32

9 c = 94

10 Eccentricity e = ca = (94)(154) = 3 / 5

11 k = (a - c) (1 + ee) = 3 / 2 × 8 / 3 = 4
 [Substitute the values of e, a and c.]

12 r = (4)(35)(1 +35cos θ) = 125 + 3cos θ.
 [The polar equation of the ellipse is r = ke1 + e cos θ.]



Hence the right answer is Option D

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