# Find the Equation to the Ellipse (Referred to Its Axes as the Axes of X and Y Respectively) Which Passes Through the Point (−3, 1) and Has Eccentricity √ 2 5 - Mathematics

Find the equation to the ellipse (referred to its axes as the axes of x and y respectively) which passes through the point (−3, 1) and has eccentricity $\sqrt{\frac{2}{5}}$

#### Solution

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

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 26 Ellipse
Exercise 26.1 | Q 4 | Page 22

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