Question

The differential equation of family of curves $x^2+y^2-2ay=0$ where ${}^{\prime }{a}^{\prime }$ is arbitrary constant is

• A. $(x^2+y^2)y_1=2xy$
• B. $2(x^2+y^2)y_1=xy$
• C. $(x^2-y^2)y_1=2xy$
• D. $2(x^2-y^2)y_1=xy$

Answer 1

The correct option is C $(x^2-y^2)y_1=2xy$

$x^2+y^2-2ay=0⟶\left(1\right)$

Differentiating the entire equation w.r.t to $x$ we have

$2x+2y\frac{dy}{dx}-2a\frac{dy}{dx}=0$

$\Rightarrow x+y\frac{dy}{dx}-a\frac{dy}{dx}=0$

$\Rightarrow \frac{dy}{dx}(y-a)=-x$

$\Rightarrow \frac{dy}{dx}=\frac{+x}{a-y}$

From $\left(1\right)$ we have $a=\frac{x^2+y^2}{2y}$

Using that we can write

$\Rightarrow \frac{dy}{dx}=\frac{x}{\frac{x^2+y^2}{2y}-y}$

$\Rightarrow \frac{dy}{dx}=\frac{2xy}{x^2-y^2}$

$\Rightarrow (x^2-y^2)y_1=2xy$ ; where $y_1=\frac{dy}{dx}$

$\therefore$ Answer : Option C.