Find the length of lactus rectum and end points of lactus rectum for a ellipse $\large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1? \begin{array}{1 1}(a)\;\large\frac{2b^2}{a}\\(b)\;\large\frac{b^2}{a}\\(c)\;\large\frac{4b^2}{a}\\(d)\;\large\frac{2a^2}{b}\end{array} 1 Answer Latus rectum is a perpendicular line passing through focus hence let the end points be L(ae,k) & L'(ae,-k). Length of latus rectum LL'=2k \large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1$
$\large\frac{a^2e^2}{a^2}+\frac{k^2}{b^2}$$=1 k^2=b^2(1-e^2) \;\;\;\;=b^2(\large\frac{b^2}{a^2}) b^2=a^2(1-e^2) k=\large\frac{b^2}{a} 2k=\large\frac{2b^2}{a}=$$LL'$(length of lactus rectum)
Hence (a) is the correct answer.
edited Sep 28, 2014