The locus of the centre of a circle that passes through the origin and cuts off a length 2a from the line y = c is

Let the locus of the center be represented by (h,k)

Since it passes through origin. Radius r = $$\sqrt{\ h^2+k^2}$$

Since it cuts off chord of length = 2a on y=c. We draw 1 perpendicular bisector of the chord till the center and make a line which is intersecting point of line y=c and the circle.

We obtain a right triangle that has "r" as its hypotenuse and a and |c-k| as its sides.

Applying Pythogoras theorm, 

$$r^2\ =\ a^2\ +\ \left(c-k\ \right)^2$$

it implies

$$h^2+k^2\ =\ a^2\ +\ \left(c-k\ \right)^2$$

$$h^2+k^2\ =\ a^2\ +\ c^2+k^2\ -2ck$$

$$h^2+2ck\ =\ a^2\ +\ c^2$$

replacing (h,k) by (x,y)

we get $$x^2+2cy\ =\ a^2\ +\ c^2$$