Form the differential equation of the family of circles touching the x-axis at origin.

Explanation

(1) Let C (0, k) be the centre of the circle touching the X - axis at the origin.
`(NVT_21_MAT_XII_C16_SLV_012_S01)`
The radius of the circle is k .
equation of the circle is ` (x-0)^(2) + (y -k)^(2) =k^(2)`
` x^(2)+y^(2) -2ky + k^(2) =k^(2)`
` x^(2) +y^(2) -2ky +k^(2)=k^(2)`
` x^(2) +y^(2) =2ky`
` x^(2)/y + y = 2k`
Wherer k is an arbitary constant.
Differentiating w.r.t we, get,
`(y.d/(dx)(x^(2))-x^(2)(dy)/(dx))/y^(2) + dy/dx = 0`
` yxx 2x -x^(2) dy/dx +y^(2)dy/dx=0 therefore 2xy = (x^(2) -y^(2)) dy/dx`
` dy/dx = (2xy)/(x^(2) -y^(2)) `
This is the required D.E.