Answer
Vertex: $(5,-3)$
Focus: $(5,\dfrac{-5}{2})$
Work Step by Step
The standard form of the equation of the circle is:
$x^2=10x -2y+19=0$
or, $(x^2-10x+25)=2y-19+25$
or, $(x-5)^2=4(1/2)(y+3)$
So, we see that the vertex is: $(5,-3)$
Focus: $(h, k+p)=(5,\dfrac{-5}{2})$
