An ellipse has its major axis along y-axis and minor axis along x-axis. If its length of latus rectum is (2/3) times of its minor axis, then the eccentricity of the ellipse is

Question

An ellipse has its major axis along y-axis and minor axis along x-axis. If its length of latus rectum is (2/3) times of its minor axis, then the eccentricity of the ellipse is (A) (2/3) (B) (3/5) (C) (√5/3) (D) (√2/5)

Tardigrade Admin · Accepted Answer

Latus rectum of ellipse =(2/3) (Minor axis) <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/fbebf1c2f2685af8327ca0f81ebcb916-.png" /> (2 a2/b)=(2/3)(2 a) ⇒ (a/b)=(2/3) a=(2/3) b ⇒ a2=b2(1-e2) (4 b2/9)=b2(1-e2) ⇒ e=(√5/3)

Q. An ellipse has its major axis along y-axis and minor axis along $x$ -axis. If its length of latus rectum is $\frac{2}{3}$ times of its minor axis, then the eccentricity of the ellipse is

$\frac{2}{3}$

$\frac{3}{5}$

$\frac{5}{3}$

$\frac{2}{5}$

Latus rectum of ellipse $= \frac{2}{3}$ (Minor axis)

$\frac{2 a ^{2}}{b} = \frac{2}{3} (2 a) \Rightarrow \frac{a}{b} = \frac{2}{3}$
$a = \frac{2}{3} b \Rightarrow a^{2} = b^{2} (1 - e^{2})$
$\frac{4 b ^{2}}{9} = b^{2} (1 - e^{2}) \Rightarrow e = \frac{5}{3}$

Q. An ellipse has its major axis along y-axis and minor axis along x-axis. If its length of latus rectum is 32​ times of its minor axis, then the eccentricity of the ellipse is

32​

53​

35​​

52​​