Find the differential equation of all the circles which pass through the origin and whose centres lie on y-axis.

Explanation

It is given that, circle pass through origin and their centresile on Y-axis. Let (0,K) be the centre of the circle and radius is K.
So, the equation of circle is
`(x-0)^(2)+(y-k)^(2)=k^(2)`
`Rightarrow x^(2)+(y-k)^(2)=k^(2)`
`Rightarrow x^(2)+y^(2)-2ky=0`
`Rightarrow ((x^(2)+y^(2)))/(2y)=k`
On differentiating Eq. (i) w.r.t. we get
`((2y)(2x+2y(dy)/(dx))-(x^(2)+y^(2))(2dy)/(dx))/(4y^(2))=0`
`Rightarrow 4y(x+y(dy)/(dx))-2(x^(2)+y^(2))(dy)/(dx)=0`
`Rightarrow 4xy+4y^(2)(dy)/(dx)-2(x^(2)+y^(2))(dy)/(dx)=0`
`Rightarrow [4y^(2)-2(x^(2)+y^(2))](dy)/(dx)+4xy=0`
`Rightarrow (4y^(2)-2x^(2)-2y^(2))(dy)/(dx)+4xy=0`
`Rightarrow (2y^(2)-2x^(2))(dy)/(dx)+4xy=0`
`Rightarrow (y^(2)-x^(2))(dy)/(dx)+2xy=0`
`Rightarrow (x^(2)-y^(2))(dy)/(dx)-2xy=0`