Question

Find the differential equations having as solutions:
all circles through (0,0) and (2,0)

Answer 1

Step-by-step explanation:

It is given that, circles pass through origin and their centres lie on Y-axis.

Let (0,k) be the centre of the circle with k as its centre.

So, the equation of circle is

(x−0)

2 +(y−k) 2 =k 2

x 2 +(y−k) 2 =k 2x 2 +y 2 −2ky=0

2yx 2 +y 2 =k .....(1)

Differentiating equation 1 w.r.t. x, we get,4y 2

2y(2x+2y dxy )−(x 2 +y 2 ) dx2dy =0

4y(x+y dxdd )−2(x 2 +y 2 ) dxdd=0

4xy=4y 2dxdd −2(x 2 +y 2 ) dxdd =0

(4y 2 −2x 2−2y 2 ) dxdd +4xy=0

(2y 2 −2x 2 )dxdd +4xy=0

(y 2 −x 2 ) dxdd +2xy=0

(x2 −y 2 ) dxdd −2xy=0

(4y 2 −2x 2 −2y 2 ) dxdd+4xy=0 (2y 2 −2x 2 ) dxdd +4xy=0

(y 2 −x 2 ) dxdd+2xy=0 (x 2 −y 2 ) dxdd −2xy=0