Form the Differential Equation of the Family of Ellipses Having Foci on Y-axis and Centre at the Origin. - Mathematics

Sum

Form the differential equation of the family of ellipses having foci on y-axis and centre at the origin.

Solution

The equation of the ellipses having foci on y-axis and centre at the origin is given by

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1............(1)$

Here,

b > a

Since these are two parameters, so we differentiate the equation twice.

Differentiating with respect to x, we get

$\frac{2x}{a^2} + \frac{2y}{b^2}y' = 0$
$\Rightarrow \frac{x}{a^2} + \frac{y}{b^2}y' = 0 . . . . . \left( 2 \right)$
$\Rightarrow \frac{1}{a^2} + \frac{1}{b^2} \left( y' \right)^2 + \frac{y}{b^2}y'' = 0 . . . . . \left( 3 \right)$
Multiplying throughout by x, we get
$\frac{x}{a^2} + \frac{x}{b^2} \left( y' \right)^2 + \frac{xy}{b^2}y'' = 0 . . . . . \left( 4 \right)$
Subtracting (2) from (4), we get
$\frac{1}{b^2}\left[ x \left( y' \right)^2 + xyy'' - yy' \right] = 0$
$\Rightarrow x \left( y' \right)^2 + xyy'' - yy' = 0$

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APPEARS IN

RD Sharma Class 12 Maths
Chapter 22 Differential Equations
Revision Exercise | Q 9 | Page 145