Equation of ellipse with eccentricity frac{1}{2} and foci at ( pm 1,0) is

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10-2. Parabola, Ellipse, Hyperbola

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Equation of the ellipse with eccentricity $\frac{1}{2}$ and foci at $( \pm 1,\;0)$ is

A

$\frac{{{x^2}}}{3} + \frac{{{y^2}}}{4} = 1$

B

$\frac{{{x^2}}}{4} + \frac{{{y^2}}}{3} = 1$

C

$\frac{{{x^2}}}{3} + \frac{{{y^2}}}{4} = \frac{4}{3}$

D

None of these

Solution

(b) Given that, $e = \frac{1}{2}$ and $( \pm \,ae,\,0) = \,( \pm \,1,\,0)$

$ \Rightarrow \,ae = 1$

$ \Rightarrow a = 2$. Now ${b^2} = {a^2}(1 – {e^2})$

$ \Rightarrow \,\,\,{b^2} = 4\left( {1 – \frac{1}{4}} \right)$

$ \Rightarrow \,\,{b^2} = 3$

Hence, equation of ellipse is $\frac{{{x^2}}}{4} + \frac{{{y^2}}}{3} = 1.$

Standard 11

Mathematics

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