Question 1062324
a. {{{x^2 - 10x + y^2 + 4y + 13 = 0}}}
{{{x^2 - 10x + y^2 + 4y = -13}}}
Completing squares:
{{{x^2 - 10x + red(25) + y^2 + 4y + green(4) = -13+red(25)+green(4)}}}
{{{(x^2 - 10x + 25) + (y^2 + 4y +4) = 16}}}
{{{highlight((x-5)^2 + (y+2)^2 = 16)}}} or {{{highlight((x-5)^2 + (y+2)^2 = 4^2)}}}
 
b. The radius is {{{highlight(4)}}} ,
the number squared on the right side of the equation above.
The coordinates of the center are the solution for the shrunken circle with {{{radius=0}}} ,
{{{(x-5)^2 + (y+2)^2 = 0}}} --> {{{system(x-5=0,y+2=0)}}} --> {{{highlight(system(x=5,y=-2))}}}  
 
c. The x-intercepts:
The x-intercepts are the points (if any) where {{{y=0}}} ,
where the graph of the equation crosses the x-axis.
To find them, we solve
{{{system(y=0,x^2 - 10x + y^2 + 4y + 13 = 0)}}}-->{{{system(y=0,x^2 - 10x + 13 = 0)}}}-->{{{"..."}}}-->{{{highlight(system(y=0,x=5 +- 2sqrt(3)))}}} ,
or we solve
{{{system(y=0,(x-5)^2 + (y+2)^2=16)}}}-->{{{system(y=0,(x-5)^2+2^2=16)}}}-->{{{system(y=0,(x-5)^2+4=16)}}}-->{{{system(y=0,(x-5)^2=16-4)}}}-->{{{system(y=0,(x-5)^2=12)}}}-->{{{system(y=0,x=5 +- sqrt(12))}}}-->{{{highlight(system(y=0,x=5 +- 2sqrt(3)))}}} .
Theapprocimate values for the y=coordinates of those points are
{{{highlight(y=about1.536)}}} and {{{highlight(y=about8.464)}}} .
 
The y-intercepts:
The y-intercepts are the points (if any) where {{{x=0}}} ,
where the graph of the equation crosses the x-axis.
To find them, we solve
{{{system(x=0,x^2 - 10x + y^2 + 4y + 13 = 0)}}}-->{{{system(x=0,y^2 + 4y + 13 = 0)}}}-->{{{"..."}}}--> {{{no}}} {{{solution}}} ,
or we solve
{{{system(x=0,(x-5)^2 + (y+2)^2 = 16)}}}-->{{{system(x=0,25+(y+2)^2=16)}}}-->{{{system(x=0,(y+2)^2=16-25)}}}-->{{{system(x=0,(y+2)^2=-9)}}}--> {{{no}}} {{{solution}}} >
The graph does not cross the y-axis.
In fact, it looks like this:
{{{drawing(300,300,-1,9,-7,3,
grid(1),
red(circle(5,-2,0.1)),
red(circle(5,-2,4))
)}}}