A circle C has equation $$x^{2} + y^{2} - 4x + 10y = k$$ where k is a constant - Edexcel - A-Level Maths Pure - Question 5 - 2017 - Paper 1

To find the center of the circle represented by the equation, we will complete the square for both the x and y terms in the equation.

Starting with the equation:
$x^{2} + y^{2} - 4x + 10y = k$

1. Completing the square for x:

   $x^{2} - 4x = (x - 2)^{2} - 4$

2. Completing the square for y:

   $y^{2} + 10y = (y + 5)^{2} - 25$

Now substitute these back into the equation:

$(x - 2)^{2} - 4 + (y + 5)^{2} - 25 = k$

This simplifies to:

$(x - 2)^{2} + (y + 5)^{2} = k + 29$

From the standard equation of a circle $(x - h)^{2} + (y - k)^{2} = r^{2}$, we can identify that the center coordinates are:

Center: (2, -5)