Question

Find the center and radius of the circle $$x^{2}+y^{2}-10 x+4 y+20=0$$

Short answer

The center is (5, -2) and the radius is 3.

Step-by-step solution

Rewrite the equation

Start with rewriting the given equation of the circle which is \(x^{2}+y^{2}-10 x+4 y+20=0\) .

Group the x and y terms

Group the x terms and y terms together. The equation becomes \(x^{2} - 10x + y^{2} + 4y + 20 = 0\) .

Move the constant term to the other side

Subtract 20 from both sides of the equation to move the constant term: \(x^{2} - 10x + y^{2} + 4y = -20\) .

Complete the square for the x terms

To complete the square for the x terms, take half the coefficient of x and square it. Half of -10 is -5, and \((-5)^{2} = 25\) . Add and subtract 25 inside the equation: \(x^{2} - 10x + 25 + y^{2} + 4y = -20 + 25\).

Complete the square for the y terms

To complete the square for the y terms, take half the coefficient of y and square it. Half of 4 is 2, and \((2)^{2} = 4\) . Add and subtract 4 inside the equation: \(x^{2} - 10x + 25 + y^{2} + 4y + 4 = -20 + 25 + 4\).

Rewrite as square of binomials

Rewrite the perfect squares as the square of binomials: \((x - 5)^{2} + (y + 2)^{2} = 9\).

Identify the center and radius

Compare the resulting equation with the standard circle equation \((x-h)^{2} + (y-k)^{2} = r^{2}\). The center (h, k) is (5, -2), and the radius r is 3.

Key concepts

completing the square

Completing the square is a technique used to transform a quadratic equation into a perfect square trinomial by adding and subtracting a specific value. This is particularly useful in circle equations to rewrite them in their standard form. To complete the square for a term involving `x`, follow these steps:

• Take half of the coefficient of `x`.
• Square that result.
• Add and subtract this square value in the equation.

For example, in the equation \(x^2 - 10x + y^2 + 4y + 20 = 0\), we complete the square for `x` by taking half of `-10` (which is `-5`), squaring it to get `25`, and then adding and subtracting `25`:
\(x^2 - 10x + 25 + y^2 + 4y = -20 + 25\)
Similarly, for `y`, take half of `4` (which is `2`), square it to get `4`, and add and subtract `4`:
\(x^2 - 10x + 25 + y^2 + 4y + 4 = -20 + 25 + 4\).
After completing the square for both variables, you'll be able to rewrite the equation as the square of binomials.

standard form of a circle

The standard form of a circle's equation allows us to easily identify its key attributes, the center, and the radius. This form is: \((x - h)^2 + (y - k)^2 = r^2\). Here, \((h, k)\) is the center of the circle, and \(r\) is its radius.
Aim to convert the given circle equation into this format through the process of completing the square.
Once we convert \(x^2+y^2-10x+4y+20=0\) using the steps of completing the square, we get the equation in the form \((x - 5)^2 + (y + 2)^2 = 9\).
Comparing it with the standard form, our equation now clearly presents that:

• The center of the circle is at \((h, k) = (5, -2)\).
• The squared radius \(r^2 = 9\), thus the radius \(r = \sqrt{9} = 3\).

radius and center of a circle

Identifying the radius and center of a circle is essential for understanding its geometry and for graphing it accurately. Once we have the equation in the standard form \((x - h)^2 + (y - k)^2 = r^2\), it's straightforward to extract these key pieces of information.
In our example, converting the given equation we found: \((x - 5)^2 + (y + 2)^2 = 9\).
From this form:

• The center \(h, k\) is \((5, -2)\).
• The term \(r^2 = 9\) indicates that \(r = 3\).

Knowing how to derive the center and radius aids in sketching the circle on a coordinate plane and better understanding the circle's properties and position.