Find the length of the major axis of an ellipse whose axis are the co-ordinate axis with latus rectum 20 , whose minor axis is the distance between the foci.

Dear Student,

Please find below the solution to the asked query :
$$\begin{gathered} \mathrm{Let} \mathrm{the} \mathrm{ellipse} \mathrm{be} \frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1 \\ \mathrm{Minor} \mathrm{axis} = 2\mathrm{b} = 2\mathrm{ae} \\ \mathrm{b}= \mathrm{ae} ......\left(1\right) \\ \mathrm{and} \mathrm{latus} \mathrm{rectum} = \frac{2{\mathrm{b}}^2}{\mathrm{a}}= 20 .....\left(2\right) \\ \mathrm{and} \mathrm{we} \mathrm{know} \mathrm{that} {\mathrm{b}}^2= {\mathrm{a}}^2\left(1-{\mathrm{e}}^2\right) \\ \frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}= 1-\frac{{\mathrm{b}}^2}{{\mathrm{a}}^2} \left[\mathrm{Using} \mathrm{equation} 1\right] \\ \frac{2{\mathrm{b}}^2}{{\mathrm{a}}^2}= 1....\left(3\right) \\ \mathrm{Using} \mathrm{equation} 2 \mathrm{and} 3 \\ 20\mathrm{a}= {\mathrm{a}}^2 \\ \mathrm{a} = 20 \\ \mathrm{So} \mathrm{b} = 10\sqrt{2} \\ \mathrm{So} \mathrm{length} \mathrm{of} \mathrm{major} \mathrm{axis}= 2\mathrm{a} = 40 \mathrm{unit} \end{gathered}$$


Hope this would clear your doubt about the topic.

If you have any more doubts just ask here on the forum and our experts will try to help you out as soon as possible.

Regards