# The differential equation of all circles passing through the origin and having their centres on the $\dpi{100} x$-axis is Option 1) $y^{2}=x^{2}+2xy\frac{dy}{dx}$ Option 2) $y^{2}=x^{2}-2xy\frac{dy}{dx}$ Option 3) $x^{2}=y^{2}+xy\frac{dy}{dx}$ Option 4) $x^{2}=y^{2}+3xy\frac{dy}{dx}$

S Sabhrant Ambastha

As we learnt in

Formation of Differential Equations -

A differential equation can be derived from its equation by the process of differentiation and other algebraical process of elimination

-

Let the equation of circle is

$(x-a)^{2}+y^{2}=a^{2}$ where (a,0) is center and a is radius.

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

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

$\Rightarrow x+yy'=a$

$\Rightarrow (yy')^{2}+ y^{2}= (x+yy')^{2}= x^{2}+ (yy')^{2}+2xyy'$

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

Option 1)

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

Correct option

Option 2)

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

Incorrect option

Option 3)

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

Incorrect option

Option 4)

$x^{2}=y^{2}+3xy\frac{dy}{dx}$

Incorrect option

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