Question

what is the radius of the circle with an equation of ( x^2 + y^2 - 2x - 8y - 7 = 0 )? a. ( sqrt{7} ) b. ( sqrt{10} ) c. ( sqrt{17} ) d. ( sqrt{24} )

Explanation:

Step1: Recall the standard circle equation

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 rewrite the given equation \(x^2 + y^2 - 2x - 8y - 7 = 0\) into this form by completing the square.

Step2: Group \(x\) and \(y\) terms

Group the \(x\)-terms and \(y\)-terms together: \((x^2 - 2x) + (y^2 - 8y) = 7\).

Step3: Complete the square for \(x\)

For the \(x\)-terms: \(x^2 - 2x\). Take half of the coefficient of \(x\) (which is \(-2\)), so \(\frac{-2}{2} = -1\), square it: \((-1)^2 = 1\). Add 1 to both sides.

Step4: Complete the square for \(y\)

For the \(y\)-terms: \(y^2 - 8y\). Take half of the coefficient of \(y\) (which is \(-8\)), so \(\frac{-8}{2} = -4\), square it: \((-4)^2 = 16\). Add 16 to both sides.

Step5: Rewrite the equation

After adding 1 and 16 to both sides, we get: \((x^2 - 2x + 1) + (y^2 - 8y + 16) = 7 + 1 + 16\).

Simplify the left side as perfect squares and the right side: \((x - 1)^2 + (y - 4)^2 = 24\)? Wait, no, wait: \(7 + 1 + 16 = 24\)? Wait, no, wait, let's recalculate: \(7 + 1 + 16 = 24\)? Wait, no, wait, the standard form is \((x - h)^2 + (y - k)^2 = r^2\), so \(r^2 = 24\)? Wait, no, wait, I must have made a mistake. Wait, let's do it again.

Wait, the original equation is \(x^2 + y^2 - 2x - 8y - 7 = 0\). So moving the constant to the right: \(x^2 + y^2 - 2x - 8y = 7\).

Completing the square for \(x\): \(x^2 - 2x\). The coefficient of \(x\) is \(-2\), half of that is \(-1\), square is \(1\). So add \(1\) to both sides.

Completing the square for \(y\): \(y^2 - 8y\). The coefficient of \(y\) is \(-8\), half of that is \(-4\), square is \(16\). So add \(16\) to both sides.

So now: \((x^2 - 2x + 1) + (y^2 - 8y + 16) = 7 + 1 + 16\).

Simplify left side: \((x - 1)^2 + (y - 4)^2\). Right side: \(7 + 1 + 16 = 24\)? Wait, but the options are \(\sqrt{7}\), \(\sqrt{10}\), \(\sqrt{17}\), \(\sqrt{24}\). Wait, but let's check again. Wait, maybe I miscalculated the right side. Wait, \(7 + 1 + 16 = 24\), so \(r^2 = 24\), so \(r = \sqrt{24}\)? But wait, let's check the options. Option D is \(\sqrt{24}\). But wait, maybe I made a mistake. Wait, let's re-express the equation.

Wait, the standard form is \((x - h)^2 + (y - k)^2 = r^2\). So if we have \((x - 1)^2 + (y - 4)^2 = 24\), then \(r^2 = 24\), so \(r = \sqrt{24}\). So the answer should be D. \(\sqrt{24}\).

Wait, but let's check again. Let's expand \((x - 1)^2 + (y - 4)^2 = 24\). Expanding: \(x^2 - 2x + 1 + y^2 - 8y + 16 = 24\). Combine like terms: \(x^2 + y^2 - 2x - 8y + 17 = 24\). Then subtract 24: \(x^2 + y^2 - 2x - 8y - 7 = 0\), which matches the original equation. So yes, \(r^2 = 24\), so \(r = \sqrt{24}\).

Answer:

D. \(\sqrt{24}\)