Calculate the center of the circle by completing the square of the equation x2+y2+2x−20y+5=0

Question

𝑥
2
+
𝑦
2
+
2
𝑥
−
20
𝑦
+
5
=
0
.

Answer 1

To find the center of the circle given by the equation

\[ x^2 + y^2 + 2x - 20y + 5 = 0, \]

we need to complete the square for both \(x\) and \(y\).

Step 1: Rearranging the equation

First, we can move the constant term to the other side:

\[ x^2 + 2x + y^2 - 20y + 5 = 0 \implies x^2 + 2x + y^2 - 20y = -5. \]

Step 2: Completing the square for \(x\)

For the \(x\) terms \(x^2 + 2x\):

1. Take the coefficient of \(x\) (which is 2), divide by 2 to get 1, and square it (which gives \(1^2 = 1\)).
2. Add this square inside the equation and balance it by subtracting the same value.

So we have:

\[ x^2 + 2x = (x + 1)^2 - 1. \]

Step 3: Completing the square for \(y\)

For the \(y\) terms \(y^2 - 20y\):

1. Take the coefficient of \(y\) (which is -20), divide by 2 to get -10, and square it (which gives \((-10)^2 = 100\)).
2. Add this square inside the equation and balance it by subtracting the same value.

So we have:

\[ y^2 - 20y = (y - 10)^2 - 100. \]

Step 4: Substitute back into the equation

Now we substitute these completed squares back into the equation:

\[ (x + 1)^2 - 1 + (y - 10)^2 - 100 = -5. \]

This simplifies to:

\[ (x + 1)^2 + (y - 10)^2 - 101 = -5. \]

Adding 101 to both sides gives:

\[ (x + 1)^2 + (y - 10)^2 = 96. \]

Step 5: Identifying the center and radius

Now, the equation is in the standard form of a circle, \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

From the equation:

\[ (x + 1)^2 + (y - 10)^2 = 96, \]

we see:

• The center \((h, k)\) is \((-1, 10)\).
• The radius is \(\sqrt{96}\).

Conclusion

The center of the circle is at

\[ \boxed{(-1, 10)}. \]