Question

Arrange the eccentricities of the following ellipses in particular order.

A- Ellipse whose Latus rectum is half of its major axis.

B- Ellipse whose distance between the foci is equal to the distance between a focus and one end of minor axis

C- Ellipse whose major axis is double the minor axis.

D- Ellipse whose distance between the foci is $6$ and the length of minor axis is $8$.

• A.
  $\left(A\right)=\left(C\right)>\left(D\right)>\left(B\right)$
  
• B.
  $\left(A\right)=\left(C\right)<\left(D\right)<\left(B\right)$
  
• C.
  $\left(A\right)>\left(C\right)>\left(D\right)>\left(B\right)$
  
• D. none of these

Answer 1

The correct option is B

$\left(A\right)=\left(C\right)<\left(D\right)<\left(B\right)$
A) Latusrectum is $\frac{2b^2}{a}$=$\frac{a}{2}$

Using $e=\sqrt{1-\frac{b^2}{a^2}}$ and above relation we get $e=\frac{\sqrt{3}}{2}$

B) Distance between foci is given by $2ae$ and this is equal to $a-ae$

This gives $e=\frac{1}{3}$

C) $e=\sqrt{1-\frac{b^2}{a^2}}$, now if $a=2b$ then $e=\frac{\sqrt{3}}{2}$

D) Given $2a=8$ and $2ae=6$ we get $e=\frac{3}{4}$

$\left(A\right)=\left(C\right)>\left(D\right)>\left(B\right)$