If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is :
Solution
Distance between foci = 2ae = 6
<br><br>$\Rightarrow$ ae = 3 .....(1)
<br><br>Distance between directrices = ${{2a} \over e}$ = 12
<br><br>$\Rightarrow$ ${a \over e}$ = 6 .....(2)
<br><br>from (1) and (2)
<br><br>a<sup>2</sup> = 18
<br><br>also a<sup>2</sup>e<sup>2</sup> = 9
<br><br>$\Rightarrow$ 18e<sup>2</sup> = 9
<br><br>$\Rightarrow$ e<sup>2</sup> = ${1 \over 2}$
<br><br>We know e<sup>2</sup> = 1 - ${{{b^2}} \over {{a^2}}}$
<br><br>$\therefore$ ${1 \over 2}$ = 1 - ${{{b^2}} \over {{a^2}}}$
<br><br>$\Rightarrow$ b<sup>2</sup> = 9
<br><br>$\therefore$ Length of latus rectum = ${{2{b^2}} \over a}$
<br><br>= ${{2 \times 9} \over {\sqrt {18} }}$
<br><br>= $3\sqrt 2$
About this question
Subject: Mathematics · Chapter: Conic Sections · Topic: Parabola
This question is part of PrepWiser's free JEE Main question bank. 146 more solved questions on Conic Sections are available — start with the harder ones if your accuracy is >70%.