The equation of any member of the family of a | vedclass

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(A) The general equation of an ellipse with axes along the coordinate axes is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Differentiating with respect to $x$

Vedclass · Accepted Answer

$x y y^{\prime \prime}+x\left(y^{\prime}\right)^2-y y^{\prime}=0$

Vedclass · Answer

(A) The general equation of an ellipse with axes along the coordinate axes is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Differentiating with respect to $x$: $\frac{2x}{a^2} + \frac{2y}{b^2} \cdot y^{\prime} = 0$ $\frac{y}{b^2} \cdot y^{\prime} = -\frac{x}{a^2}$ $\frac{y y^{\prime}}{x} = -\frac{b^2}{a^2}$ Differentiating again with respect to $x$: $\frac{d}{dx} \left( \frac{y y^{\prime}}{x} \right) = \frac{d}{dx} \left( -\frac{b^2}{a^2} \right) = 0$ Using the quotient rule: $\frac{x(y y^{\prime \prime} + (y^{\prime})^2) - y y^{\prime}(1)}{x^2} = 0$ $x y y^{\prime \prime} + x(y^{\prime})^2 - y y^{\prime} = 0$. Thus, option $A$ is correct.

The equation of any member of the family of all the ellipses whose axes are along the coordinate axes satisfies the differential equation

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