1. Tardigrade
  2. Question
  3. Mathematics
  4. An ellipse has foci (4,2),(2,2) and it passes through the point P(2,4) . The eccentricity of the ellipse is

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. An ellipse has foci $\left(4,2\right),\left(2,2\right)$ and it passes through the point $P\left(2,4\right)$ . The eccentricity of the ellipse is

NTA AbhyasNTA Abhyas 2020Conic Sections

Solution:

Let, $\left(4 , 2\right)=S_{1}$ and $\left(2 , 2\right)=S_{2}$ and eccentricity of ellipse is $e$
then $S_{1}S_{2}=2ae$ and $P S_{1}+P S_{2}=2a$
(where $2a$ is length of major axis)
$\Rightarrow e=\frac{S_{1} S_{2}}{P S_{1} + P S_{2}}=\frac{2}{2 \sqrt{2} + 2}$
$\Rightarrow e=\frac{1}{\sqrt{2} + 1}=\sqrt{2}-1=tan\frac{\pi }{8}$