QJEEM18Ma9aa8: JEE Main 2018 Mathematics - Class 11 JEEM18Ma9aa8 Solution: If the length of the latus rectum of an ellipse is 4 units and…

If the cordinate of focus and vertex are (ae, 0) and (a, 0) respectively,

then distance between focus and vertex,

a $-$ ae = ${3 \over 2}$ (given)

$\Rightarrow$ $\,\,\,$ a (1 $-$ e) = ${3 \over 2}$

Length of latus rectum,

${{2{b^2}} \over a} = 4$

$\Rightarrow$ $\,\,\,$ b² = 2a

$\Rightarrow$ $\,\,\,$ a²(1 $-$ e²) = 2a   [As b² = a² (1 $-$ e²)]

$\Rightarrow$ $\,\,\,$ a (1 $-$ e) ( 1 + e) = 2

Putting    a (1 $-$ e) = ${3 \over 2}$

$\Rightarrow$ $\,\,\,$ ${3 \over 2}$ (1 + e) = 2

$\Rightarrow$ $\,\,\,$ 3 + 3e = 4

$\Rightarrow$ $\,\,\,$ e = ${1 \over 3}$