Given you have the equation of a circle and a point inside it, how do you go about calculating the equation of a chord that has that point as its midpoint?

I have figured that i will need to find the two points on the circumfrence whose x values and y values average to the midpoint. Getting the equation should then be trivial.

To give an example: let circle C be x^2 + y^2 -2x + 4y -11 = 0

Let the desired midpoint,A, be (2,-1)

So x2-x1 = 4 and y2-y1 = -2 (with regard to the points i am looking for on the circumfrence)

and x1 and y1 and x2 and y2 must fit into the equation C.

From here i'm not sure how to solve this problem simultaniously. Is my inital work correct?

Thanks in advance