First slide

Ellipse

Question

Equation of the circle passing through the intersection of ellipses $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and $\frac{x^{2}}{b^{2}} + \frac{y^{2}}{a^{2}} = 1$ is

Moderate

A

$x^{2} + y^{2} = a^{2}$
B

$x^{2} + y^{2} = b^{2}$
C

$x^{2} + y^{2} = \frac{a^{2} b^{2}}{a^{2} + b^{2}}$
D

$x^{2} + y^{2} = \frac{2 a^{2} b^{2}}{a^{2} + b^{2}}$

Solution

Given equations of ellipse are
$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and $\frac{x^{2}}{b^{2}} + \frac{y^{2}}{a^{2}} = 1$
The point of the intersection of these ellipse are
$\begin{aligned} (\pm \frac{ab}{\sqrt{a^{2} + b^{2}}}, \pm \frac{ab}{\sqrt{a^{2} + b^{2}}}) \\ P (\frac{ab}{\sqrt{a^{2} + b^{2}}}, \frac{ab}{\sqrt{a^{2} + b^{2}}}) \end{aligned}$
The distance between OP= r
$\begin{array}{l} = \sqrt{{(\frac{ab}{\sqrt{a^{2} + b^{2}}} - 0)}^{2} + {(\frac{ab}{\sqrt{a^{2} + b^{2}}} - 0)}^{2}} \\ = \frac{ab}{\sqrt{a^{2} + b^{2}}} \sqrt{2} \end{array}$
$∴$ Equation of circle is $x^{2} + y^{4} = r^{2} \Rightarrow x^{2} + y^{2} = \frac{2 a^{2} b^{2}}{a^{2} + b^{2}}$

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