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

$\begin{array}{1 1}(A)xyy'' -x(y')^2 - yy' = 0 \\(B)\;xyy'' + x(y')^2 - yy' = 0 \\(C)\;xyy'' + x(y')^2 + yy' = 0 \\(D)\;xyy'' + x(y')^2 +y' = 0 \end{array}$

• Hence the equation of the ellipse is $\large\frac{x^2}{b^2} +\frac{ y^2}{a^2 }$$= 1 Step 1: Using the information from the tool box we write the equation as \large\frac{x^2}{b^2} +\frac{ y^2}{a^2}$$ = 1$--------(1)
$\large\frac{2x}{b^2} + \frac{2yy'}{b^2}$$= 0--------(2) Step 2: Again differentiating this we get, \large\frac{1}{b^2} + (\frac{1}{a^2})$$[yy'' +(y')^2] = 0$
$\large\frac{1}{b^2} = - [\frac{1}{a^2}]$$(yy'' + (y')^2 Substituting this in equation (2) we get, -[\large\frac{x}{a^2}]$$[yy'' + (y')^2] +\large\frac{ yy'}{a^2}$$= 0$
$xyy'' + x(y')^2 - yy' = 0$