Answer
[tex](x+5)^2+(y+6)^2=9[/tex]
Explanation:
The given equation for a circle is:
[tex]x^2+10x+y^2+12y+52=0[/tex]
Re-arrange the equation:
[tex]x^2+y^2+10x+12y+52=0----i[/tex]
The equation of a circle center of coordinates, (a, b) and radius, r is given by:
[tex]x^2+y^2-2ax-2by+a^2+b^2-r^2=0----ii[/tex]
Comparing (i) and (ii)
[tex]\begin{gathered} -2a=10 \\ a=-\frac{10}{2} \\ a=-5 \\ Also, \\ -2b=12 \\ b=-\frac{12}{2} \\ b=-6 \\ \text{And, a}^2+b^2-r^2=52 \\ (-5)^2+(-6)^2-r^2=52 \\ 25+36-r^2=52 \\ 61-r^2=52 \\ r^2=61-52 \\ r^2=9 \end{gathered}[/tex]
The equation of a circle in standard form is given by:
[tex](x-a)^2+(y-b)^2=r^2[/tex]
Substitute a = -5, b = -6 and r² = 9 into the equation of a circle in standard form above:
[tex]\begin{gathered} (x--5)^2+(y--6)^2=9 \\ \Rightarrow(x+5)^2+(y+6)^2=9 \end{gathered}[/tex]
