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 minimize the distance between the points P and Q, we need to find the value of (x, y) that minimizes the function representing the square of the distance between the points.

Let's denote the coordinates of point P as (p, q) and the coordinates of point Q as (x, y). The formula for the distance between two points in a plane is given by the distance formula:

Distance(PQ) = √((x - p)^2 + (y - q)^2)

Since we want to minimize the distance, we can minimize the square of the distance, which simplifies the calculations. Therefore, we can minimize the function:

f(x, y) = (x - p)^2 + (y - q)^2

Now, let's substitute the equations of the given graphs into the function f(x, y). For the equation of the graph containing point P:

x^2 + y^2 - 136x + 12y + 4560 = 0

And for the equation of the graph containing point Q:

(y + 6)^2 = x^3 - 116x^2 - 417x + 267460

Let's solve these equations simultaneously to find the values of p, q, x, and y.

First, solve the equation of the graph containing point P for y:

y = -x^2 + 136x - 12y - 4560

Rearrange the equation:

13y = -x^2 + 136x - 4560

Divide by 13:

y = (-1/13)x^2 + (136/13)x - 352

Now, substitute this equation for y in the equation of the graph containing point Q:

((-1/13)x^2 + (136/13)x - 352 + 6)^2 = x^3 - 116x^2 - 417x + 267460

Simplify and expand the equation:

((-1/13)x^2 + (136/13)x - 346)^2 = x^3 - 116x^2 - 417x + 267460

Now, we have a single equation involving only x. We can expand and simplify this equation to find the equation containing x only after minimization has occurred.

Once we find the values of x and y that satisfy this equation, we can substitute them into the equation for the graph containing point P to find the specific equation that represents the coordinates of point P after minimization has occurred.

Note that finding an exact equation for the minimized distance involves solving a complex equation, which may require numerical methods or approximations.