Question

the eccentricity of an ellipse with its centre at the origin is 1/2.if one of the directrix is x=4,then equation of ellipse is

Answer 1

Answer:

The equation of the required ellipse is

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

Step-by-step explanation:

Given:

$\text{Eccentricity,\;e=}\frac{1}{2}$

$\text{As per given data, the major axis of the elliopse is along x axis}$

$\text{Equauation of directrix is }x=\frac{a}{e}$

$\implies\frac{a}{e}=4$

$\implies\frac{a}{\frac{1}{2}}=4$

$\implies\;2a=4$

$\implies\bf\;a=2$

$\text{Also}$

$b^2=a^2(1-e^2)$

$b^2=4(1-\frac{1}{4})$

$b^2=4(\frac{3}{4})$

$b^2=3$

$\text{since the major axis is along x axis and the centre is at origin, }$

$\text{the equation of the ellipse is }\\\\\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

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