The equation of the circle is $x^{2} + y^{2} - 12 x + 6 y + 20 = 0$ .
What is the radius of the circle?
Enter your answer in the box. $r =$ ___ units.

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Solution

Step-1: The general form of the equation of the circle:
The general form of the equation of the circle is ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ ,
where $(h, k)$ is the center of the circle and the $r$ is the radius.
Step 2: To convert the given equation into the standard form.
Given equation is $x^{2} + y^{2} - 12 x + 6 y + 20 = 0$ .
Rewrite the given equation as
$x^{2} - 12 x + y^{2} + 6 y = - 20$
Add $36$ and $9$ on both sides of the equation.
$x^{2} - 12 x + 36 + y^{2} + 6 y + 9 = - 20 + 36 + 9 \Rightarrow x^{2} + 2 (- 6) x + {(- 6)}^{2} + y^{2} + 2 (3) + 3^{2} = 25 ∴ {(x - 6)}^{2} + {(y + 3)}^{2} = 5^{2}$
By comparing it with the standard equation we obtain $r = 5$ .
Hence, the value of $r$ is $5$ units.

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