Question

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

Answer 1

To find the radius of the circle defined by the equation \( x^2 + y^2 - 14x - 4y = 11 \), we'll first rearrange the equation by completing the square for both the \( x \) and \( y \) terms.

### Step 1: Rearranging the Equation
Start with the original equation:
\[
x^2 + y^2 - 14x - 4y = 11.
\]

### Step 2: Completing the Square for \( x \)
Focus on the \( x \) terms:
\[
x^2 - 14x.
\]
To complete the square, take half of \(-14\), which is \(-7\), and square it:
\[
\left(-7\right)^2 = 49.
\]
Thus, we write:
\[
x^2 - 14x = (x - 7)^2 - 49.
\]

### Step 3: Completing the Square for \( y \)
Now, for the \( y \) terms:
\[
y^2 - 4y.
\]
Take half of \(-4\), which is \(-2\), and square it:
\[
\left(-2\right)^2 = 4.
\]
Thus, we write:
\[
y^2 - 4y = (y - 2)^2 - 4.
\]

### Step 4: Substitute Back into the Equation
Incorporating these completed squares back into the original equation gives:
\[
((x - 7)^2 - 49) + ((y - 2)^2 - 4) = 11.
\]
Simplifying this, we have:
\[
(x - 7)^2 + (y - 2)^2 - 53 = 11.
\]

### Step 5: Move Constant to the Other Side
Now we'll move \(-53\) to the other side:
\[
(x - 7)^2 + (y - 2)^2 = 11 + 53.
\]
This simplifies to:
\[
(x - 7)^2 + (y - 2)^2 = 64.
\]

### Step 6: Identify the Center and Radius
The equation \((x - 7)^2 + (y - 2)^2 = 64\) is in standard form for a circle, where the center is \((7, 2)\) and the radius \(r\) is given by:
\[
r = \sqrt{64}.
\]
Thus,
\[
r = 8.
\]

### Conclusion
The radius of the circle is \(\boxed{8}\).