Answer :
The equation of the circle given by the problem is:
[tex]x^2-4x+y^2-10y=-4[/tex]Step to find the radius:
Step 1. Rearrange the terms from the greatest exponent to the lowest exponent:
[tex]x^2+y^2-4x-10y=-4[/tex]Step 2. We need for this equation to be equal to 0, so we add 4 to both sides:
[tex]\begin{gathered} x^2+y^2-4x-10y+4=-4+4 \\ x^2+y^2-4x-10y+4=0 \end{gathered}[/tex]Step 3. Compare the last equation with the general equation:
[tex]x^2+y^2+Dx+Ey+F=0[/tex]Comparing the equations we find the values for D, E, and F:
[tex]\begin{gathered} D=-4 \\ E=-10 \\ F=4 \end{gathered}[/tex]Step 4. With the values of D and E, find the values of a and b defined as follows:
[tex]\begin{gathered} a=-\frac{D}{2} \\ b=-\frac{E}{2} \end{gathered}[/tex]Substituting D=-4 and E=-10:
[tex]\begin{gathered} a=-\frac{(-4)}{2}=\frac{4}{2}=2 \\ b=-\frac{(-10)}{2}=\frac{10}{2}=5 \end{gathered}[/tex]a=2 and b=5.
Step 5. The formula to find the radius is:
[tex]r=\sqrt[]{a^2+b^2-F}[/tex]Substituting the values of a, b and F:
[tex]r=\sqrt[]{2^2+5^2-4}[/tex]Solving the operations:
[tex]r=\sqrt[]{4+25-4}[/tex][tex]r=\sqrt[]{25}[/tex][tex]r=5[/tex]The radius is equal to 5.
Answer: r=5
