Find the differential equation of all circles tangent to y-axis?

Answer 1

Let the circle with centre (a, b) be tangent to the y-axis.

Then, the radius of the circle must be b.

Therefore the equation of the circle is

(x - a)2 + (y - b)2 = b2

or x2 - 2ax + a2 + y2 - 2by = 0

Then 2x - 2a + 2ydy/dx - 2bdy/dx = 0

ie x - a + ydy/dx - bdy/dx

(y - b)dy/dx = a - x

so dy/dx = (a - x)/(y - b) or -(x - a)/(y - b)