- Home
- Standard 11
- Mathematics
10-2. Parabola, Ellipse, Hyperbola
medium
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 semi-major axis is
A
$\frac{8}{3}$
B
$\frac{2}{3}$
C
$\frac{4}{3}$
D
$\frac{5}{3}$
(AIEEE-2008)
Solution
Obviously the major axis is along the $x$ -axis The distance between the focus and the
corresponding directrix $=\left|\frac{a}{e}-a e\right|=4$
$\left.\Rightarrow \frac{a}{e}-a e=4 \quad \text { (note that } \frac{a}{e}>a e\right)$
$\Rightarrow a\left(\frac{1}{e}-e\right)=4 \Rightarrow a\left(2-\frac{1}{2}\right)=4$
$\Rightarrow a \cdot \frac{3}{2}=4 \therefore a=\frac{8}{3}$
Standard 11
Mathematics
