Question

Calculate the radius of the circle by completing the square of the equation x2+y2−14x−4y=11
𝑥
2
+
𝑦
2
−
14
𝑥
−
4
𝑦
=
11

Answer 1

To find the radius of the circle from the equation \( x^2 + y^2 - 14x - 4y = 11 \), we can complete the square for the \( x \) and \( y \) terms.

1. Rearranging the equation:
\[
x^2 - 14x + y^2 - 4y = 11
\]

2. Completing the square for the \( x \) terms:
\[
x^2 - 14x \quad \text{can be written as} \quad (x - 7)^2 - 49
\]
Here, \( -14x \) can be expressed as \( -2 \cdot 7x \) and \( 7^2 = 49 \).

3. Completing the square for the \( y \) terms:
\[
y^2 - 4y \quad \text{can be written as} \quad (y - 2)^2 - 4
\]
Here, \( -4y \) can be expressed as \( -2 \cdot 2y \) and \( 2^2 = 4 \).

4. Updating the original equation with completed squares:
\[
(x - 7)^2 - 49 + (y - 2)^2 - 4 = 11
\]

5. Combine constants on the right side:
\[
(x - 7)^2 + (y - 2)^2 - 53 = 11
\]
\[
(x - 7)^2 + (y - 2)^2 = 11 + 53
\]
\[
(x - 7)^2 + (y - 2)^2 = 64
\]

6. The equation \((x - 7)^2 + (y - 2)^2 = 64\) represents a circle centered at \((7, 2)\) with a radius \( r \) where:
\[
r^2 = 64
\]
Therefore:
\[
r = \sqrt{64} = 8
\]

Thus, the radius of the circle is \( \boxed{8} \).