Answer :
Given the equation of the circle:
[tex]x^2+4x+y^2-12y=-24[/tex]We will complete the squares for x and y to write the equation of the circle as:
[tex](x-h)^2+(y-k)^2=r^2[/tex]Where (h,k) is the coordinates of the circle and r is the radius of the circle
so, the equation will be:
[tex]\begin{gathered} (x^2+4x+4)+(y^2-12y+36)=-24+4+36 \\ \\ (x+2)^2+(y-6)^2=16 \\ \\ (x+2)^2+(y-6)^2=4^2 \end{gathered}[/tex]So, the radius of the circle = 4
And the center of the circle = ( -2, 6 )
Now, we will make the transformation for the circle:
Shift 1 unit up, the center will be = ( -2, 7 )
shift 2 units right, the center will be = ( 0, 7 )
Reflection across the x-axis, the center will be = ( 0, -7)
So, the equation of the circle after transformation will be:
[tex](x-0)^2+(y+7)^2=4^2[/tex]So, the answer will be:
[tex](x-0)^2+(y+7)^2=16[/tex]