Question

what is the radius of a circle whose equation is ( x^2 + y^2 - 10x + 6y + 18 = 0 )?

• 2 units
• 4 units
• 8 units
• 16 units

Explanation:

Step1: Recall circle equation form

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h,k)\) is the center and \(r\) is the radius. We need to convert the given equation \(x^2 + y^2 - 10x + 6y + 18 = 0\) to this form by completing the square.

Step2: Group x and y terms

Group the \(x\)-terms and \(y\)-terms: \((x^2 - 10x) + (y^2 + 6y) = -18\).

Step3: Complete the square for x

For \(x^2 - 10x\), take half of \(-10\) (which is \(-5\)), square it (\((-5)^2 = 25\)), and add to both sides:
\((x^2 - 10x + 25) + (y^2 + 6y) = -18 + 25\).

Step4: Complete the square for y

For \(y^2 + 6y\), take half of \(6\) (which is \(3\)), square it (\(3^2 = 9\)), and add to both sides:
\((x^2 - 10x + 25) + (y^2 + 6y + 9) = -18 + 25 + 9\).

Step5: Rewrite as perfect squares

Rewrite the grouped terms as perfect squares: \((x - 5)^2 + (y + 3)^2 = 16\).

Step6: Identify radius

Now the equation is in standard form: \((x - 5)^2 + (y + 3)^2 = 4^2\). Comparing with \((x - h)^2 + (y - k)^2 = r^2\), we see \(r^2 = 16\), so \(r = 4\) (radius is positive).

Answer:

4 units (corresponding to the option with "4 units")