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Question: The parametric representation of a point on the ellipse whose foci are (-1, 0) and (7, 0) and eccent...

The parametric representation of a point on the ellipse whose foci are (-1, 0) and (7, 0) and eccentricity is 12\frac{1}{2}, is

A

(3+8cos⁡θ,43sin⁡θ)\left( 3 + 8\cos\theta,4\sqrt{3}\sin\theta \right)

B

(8cos⁡θ,43sin⁡θ)\left( 8\cos\theta,4\sqrt{3}\sin\theta \right)

C

(3+43cos⁡θ,8sin⁡θ)\left( 3 + 4\sqrt{3}\cos\theta,8\sin\theta \right)

D

None of these

Answer

(3+8cos⁡θ,43sin⁡θ)\left( 3 + 8\cos\theta,4\sqrt{3}\sin\theta \right)

Explanation

Solution

Foci are (-1, 0) & (7, 0)

Co-ordinate of the centre is (3, 0)

Distance between foci = (−1−7)2+0=8\sqrt{( - 1 - 7)^{2} + 0} = 8

2ae = 8

e1e_{1}

Here e=12e = \frac{1}{2}

∴a=8∴b2=a2(1−e2)=64(1−14)\therefore a = 8\therefore b^{2} = a^{2}\left( 1 - e^{2} \right) = 64\left( 1 - \frac{1}{4} \right)=64−16=4864 - 16 = 48

Equation of Ellipse with centre (3, 0) is b2=16b^{2} = 16

∴\thereforeParametric co-ordinates(3+8cos⁡θ,43sin⁡θ)\left( 3 + 8\cos\theta,4\sqrt{3}\sin\theta \right)