Answer:
Explanation:
Given the equation of the circle below:
[tex]$x^{2}+y^{2}-4 x-10 y+20=0$[/tex]
We are required to find the coordinates of the center and the length of the radius.
In order to do this, we complete the squares for each of the variables x and y.
First, rearrange the equation:
[tex]x^2-4x+y^2-10y=-20[/tex]
To complete the square for x, divide the coefficient of x by 2, square it and add it to both sides of the equation.
[tex]x^2-4x+\left(-\frac{4}{2}\right)^2+y^2-10y=-20+\left(-\frac{4}{2}\right)^2[/tex]
Repeat the same process for y:
[tex]\begin{gathered} x^2-4x+\left(-\frac{4}{2}\right)^2+y^2-10y+\left(-\frac{10}{2}\right)^2=-20+\left(-\frac{4}{2}\right)^2+\left(-\frac{10}{2}\right)^2 \\ x^2-4x+(-2)^2+y^2-10y+(-5)^2=-20+(-2)^2+(-5)^2 \end{gathered}[/tex]
Write the perfect squares and simplify the right-hand side:
[tex]\begin{gathered} (x-2)^2+(y-5)^2=-20+4+25 \\ (x-2)^2+(y-5)^2=9 \end{gathered}[/tex]
Compare to the standard form of a circle below:
[tex]\begin{gathered} (x-a)^2+(y-b)^2=r^2 \\ \implies(a,b)=(2,5) \\ r^2=9\implies r=3 \end{gathered}[/tex]
• The center of the circle = (2, 5)
,
• The length of the radius = 3 units
Option B is correct.
