Confusion regarding usage of common chord formula

Question

If the circle C1 : x2 + y2 =16 intersects another circle C2 of radius 5 in such a manner that common chord is of maximum length and has a slope equal to 3/4 , then the coordinates of the centre of C2 can be. After finding the equation of common chord, why cant I use S-S'=0 and then compare with the equation of common chord to get the equation of second circle?

Answer 1

If the said common chord has to be of maximum length, it must be equal to the length of the diameter of the samller circle.

If the common chord has slope = 3/4, then that common chord must lie on $y = 3x/4$

Solving $x^2 + y^2 = 16$ and $y = 3x/4$ will give the co-ordinates of the points of intersection (X and also Y).

That common chord must perpendicular to the line of centers. In other words, the required center (K) must lie on the line $y = -4x/3$. That is, K = (h, -4h/3) for some h.

Let the equation of the second circle be $(x - h)^2 + (y - -4h/3)^2 = 5^2$. Since this circle will pass through X (and also Y), you should have enough info to determine the loaction of that center.

Note that there two possible positions for the centers.