If the differential equation representing the family of all circles touching -axis at the origin is equals : Option 1) Option 2) Option 3) Option 4)

V Vakul

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

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Let the equation of circle is

$( x-o) \right )^{2}+(y-a)^{2}=a^{2}$

$\Rightarrow \left ( x)^{2}+(y-a)^{2}=a^{2}\ \, \, \, .....(i)$

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

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

$\therefore \frac{dy}{dx}=\frac{x}{(y-a)}$      or     $a =\frac{x+y\frac{dy}{dy}}{dy/dx}$

Put a in (i)

$x^{2}+ \frac{x^{2}}{(dy/dx)^{2}} = (\frac{x+y\frac{dy}{dx}}{(dy/dx)})^{2}$

$\\ \Rightarrow (x^{2}-y^{2})\frac{dy}{dx}=2xy \\ \therefore g(x) =2x$

Option 1)

This option is incorrect.

Option 2)

This option is incorrect.

Option 3)

This option is correct.

Option 4)

This option is incorrect.

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