Question

17. Find the equation of the ellipse whose foci are (+2,0) and the eccentricity 1 is 2​

Answer 1

Step-by-step explanation:

Given foci are (±2,0) and eccentricity $\sf{e=\dfrac{1}{2}}$

Now,

$\sf{ ae = 2}$

$\sf{ \implies \: a \cdot \dfrac{1}{2} = 2}$

$\sf{ \implies \: a= 4}$

Also,

$\sf{{e}^{2} = \dfrac{1}{4}}$

$\sf{ \implies1 - \dfrac{{b}^{2} }{a^{2} }= \dfrac{1}{4}}$

$\sf{ \implies1 - \dfrac{{b}^{2} }{16 }= \dfrac{1}{4}}$

$\sf{ \implies \dfrac{16 - {b}^{2} }{16 }= \dfrac{1}{4}}$

$\sf{ \implies 16 - {b}^{2} = \dfrac{16}{4}}$

$\sf{ \implies 16 - {b}^{2} = 4}$

$\sf{ \implies {b}^{2} = 12}$

Now,

Equation of ellipse is

$\sf{\dfrac{x^{2} }{ {a}^{2} } + \dfrac{ {y}^{2} }{ {b}^{2} } = 1}$

$\sf{ \implies\dfrac{x^{2} }{16 } + \dfrac{ {y}^{2} }{ 12 } = 1}$