Well, I have a plan but you'll need to put in the details.

If the circle passes through the origin then .

Then the equation of the circle becomes

.... (1)

Differentiate (1) with respect to x:

.... (2)

Differentiate (2) with respect to x:

.... (3)

Clearly k needs to be eliminated from (3).

(1) and (2) can be solved simultaneously to get k and h in terms of x, y and y'. You want k. Substitute the result into (3).