If point P(x,y) is equidistant from point A(3,6), this means that the distance between P and A is equal to the distance between P and any other point on the same line.

Using the distance formula, the distance between two points (x1,y1) and (x2,y2) is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2)

So, the distance between P(x,y) and A(3,6) is:

d1 = √((x - 3)^2 + (y - 6)^2)

Now, let's consider another point B(a,b) on the same line. The distance between P(x,y) and B(a,b) is:

d2 = √((x - a)^2 + (y - b)^2)

Since P is equidistant from A and B, d1 = d2. Therefore, we have the equation:

√((x - 3)^2 + (y - 6)^2) = √((x - a)^2 + (y - b)^2)

To simplify the equation, we can square both sides:

(x - 3)^2 + (y - 6)^2 = (x - a)^2 + (y - b)^2

Expanding and simplifying, we get:

(x^2 - 6x + 9) + (y^2 - 12y + 36) = (x^2 - 2ax + a^2) + (y^2 - 2by + b^2)

Now, let's group the x and y terms separately:

(x^2 - x^2) + (-6x + 2ax) + (9 - a^2) + (y^2 - y^2) + (-12y + 2by) + (36 - b^2) = 0

Simplifying further, we have:

(-6 + 2a)x + (9 - a^2) + (-12 + 2b)y + (36 - b^2) = 0

This equation represents the line on which P(x,y) must lie in order to be equidistant from A(3,6).