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  • Find the parametric representation of a point on the ellipse whose foci are and <img alt="(7,0)" src="https:/

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

Option: 1

\mathrm{(8 \cos \theta, 4 \sqrt{3} \sin \theta)}


Option: 2

\mathrm{(8 \cos \theta, 48 \sin \theta)}


Option: 3

\mathrm{(3-8 \cos \theta,-4 \sqrt{3} \sin \theta)}


Option: 4

\mathrm{(3+8 \cos \theta, 4 \sqrt{3} \sin \theta)}


Answers (1)

best_answer

Foci are (-1,0) and (7,0).

Distance between foci is \mathrm{2 a e=8 \Rightarrow a e=4} and since \mathrm{e=\frac{1}{2}},

so \mathrm{a=8}

Now, \mathrm{b^2=a^2\left(1-e^2\right)}

\mathrm{ \Rightarrow b^2=48 \Rightarrow b=4 \sqrt{3} }

The centre of the ellipse is the mid point of the line joining two foci, so the coordinates of the centre are (3,0).

Hence, its equation is \mathrm{\frac{(x-3)^2}{8^2}+\frac{(y-0)^2}{(4 \sqrt{3})^2}=1}                  .....(i)

Thus, the parametric coordinates of a point on (i) are 

\mathrm{(3+8 \cos \theta, 4 \sqrt{3} \sin \theta)}

Posted by

HARSH KANKARIA

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