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

Question

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 (A) (8/3) (B) (2/3) (C) (4/3) (D) (5/3)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/32c683b5524dff5851e5180a10d56b26-.png" /> Perpendicular distance of directrix from focus =(a/e)-a e=4 ⇒ a(2-(1/2))=4 ⇒ a=(8/3) ∴ Semi-major axis =8 / 3

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

$\frac{8}{3}$

$\frac{2}{3}$

$\frac{4}{3}$

$\frac{5}{3}$

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

Q. A focus of an ellipse is at the origin. The directrix is the line x=4 and the eccentricity is 21​. Then the length of the semimajor axis is

38​

32​

34​

35​