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  4. A focus of an ellipse is at the origin. The directrix is the line x=4 and the eccentricity is (1/2). Then the length of the semimajor axis is

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Q. A focus of an ellipse is at the origin. The directrix is the line $x=4$ and the eccentricity is $\frac{1}{2}$. Then the length of the semimajor axis is

Conic Sections

Solution:

image
Perpendicular distance of directrix from focus
$=\frac{a}{e}-a e=4$
$\Rightarrow a\left(2-\frac{1}{2}\right)=4$
$\Rightarrow a=\frac{8}{3}$
$\therefore $ Semi-major axis $=8 / 3$