If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is : [2025]

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

Solution

Q.
If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is : [2025]

1 $\frac{3}{\sqrt{19}}$

2 $\frac{\sqrt{5}}{7}$

3 $\frac{\sqrt{3}}{16}$

4 $\frac{4}{\sqrt{17}}$

Ans.
(4)

Let $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ be the given ellipse with length of minor axis as 2b and distance between foci is 2ae.

$∴ 2 b = \frac{1}{4} \times (2 a e) \Rightarrow 4 b = a e \Rightarrow \frac{b}{a} = \frac{e}{4}$ ... (i)

We know, $e = \sqrt{1 - \frac{b^{2}}{a^{2}}} \Rightarrow e = \sqrt{1 - \frac{e^{2}}{16}}$ [From (i)]

$\Rightarrow e^{2} = \frac{16 - e^{2}}{16} \Rightarrow 16 e^{2} = 16 - e^{2} \Rightarrow 17 e^{2} = 16$

$e = \frac{4}{\sqrt{17}}$ .

Want Us to Email you About Special Offers & Updates?