There is a point (P) on the graph of [x^2+y^2- 136 x + 12 y + 4560 = 0] and a point(Q) on the graph of

[(y + 6)^2 = x^3 - 116 x^2 - 417 x + 267460] such that the distance between them is as small as possible.

To solve this problem, we let ((x,y) be the coordinates of the point Q. Then we need to minimize the following function of (x) and (y): What will be the equation containing x and y after minimization has occured?

To find the equation containing x and y after the minimization has occurred, let's break down the problem step by step:

Step 1: Rewrite the equations in standard form:
First, let's rewrite the equation of the graph [x^2+y^2- 136x + 12y + 4560 = 0] in standard form. By completing the square for x and y, we can rewrite it as:
(x - 68)^2 + (y + 6)^2 = 2164

Next, let's rewrite the equation of the graph [(y + 6)^2 = x^3 - 116x^2 - 417x + 267460] in standard form. By rearranging the terms, it becomes:
x^3 - 116x^2 - 417x + (y + 6)^2 - 267460 = 0

Step 2: Define the distance function:
Now, to find the distance between the points (P and Q), we can use the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

In this case, let's consider P as the point (x, y) on the first graph and Q as the point ((x, y) on the second graph. So, the distance function can be expressed as:
d = sqrt((x - x)^2 + (y - y)^2)

Simplifying this, we get:
d = sqrt((0)^2 + (y - y)^2)
d = sqrt(0 + 0)
d = 0

Therefore, the minimum distance between the points P and Q is 0.

Step 3: Equating the distance function to 0:
Now, to find the values of x and y that satisfy the minimum distance (d = 0), we can equate the distance function to 0:
0 = sqrt((x - x)^2 + (y - y)^2)

As you can see, both terms (x - x)^2 and (y - y)^2 are equal to 0, indicating that any value of x and y will satisfy the equation, as long as they are on the respective graphs.

Hence, the equation containing x and y after the minimization has occurred is 0 = 0.