Question

Find the equation of an ellipse whose axes lie along the coordinate axes, which passes through the point (−3, 1) and has eccentricity equal to \[\sqrt{2/5}\] 

Answer 1

\[\text{ Let the equation of the ellipse be }\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 ...(1)\]
\[\text{ It passes through the point }\left( -3,1 \right).\]
\[\therefore\frac{9}{a^2}+\frac{1}{b^2}=1 ...( 2)\]
\[\text{ and } e = \sqrt{\frac{2}{5}}\]
\[\text{ Now }, b^2 = a^2 \left( 1 - e^2 \right)\]
\[ \Rightarrow b^2 = a^2 \left( 1 - \frac{2}{5} \right)\]
\[ \Rightarrow b^2 = a^2 \times \frac{3}{5}or\frac{3 a^2}{5}\]
\[\text{ Substituting the value of b^2 in eq. (2), we get }:\]
\[\frac{9}{a^2}+\frac{5}{3 a^2}=1\]
\[ \Rightarrow \frac{27 + 5}{3 a^2} = 1\]
\[ \Rightarrow a^2 = \frac{32}{3}\]
\[ \Rightarrow b^2 = \frac{3 \times \frac{32}{3}}{5} or \frac{32}{5}\]
\[\text{ Substituting the values ofaandbin eq. (1), we get }:\]
\[\frac{3 x^2}{32} + \frac{5 y^2}{32} = 1\]
\[ \Rightarrow 3 x^2 + 5 y^2 = 32\]
\[\text{ This is the required equation of the ellipse }.\]