Question

The circumference of the circle \(x^2 + y^2 - 18x - 16y + 120 = 0\) is:

• A. \(5\pi\)
• B. \(10\pi\)
• C. \(25\pi\)
• D. \(10\pi^2\)

Hint:

For circle equations, completing the square is the quickest way to find the radius and thus compute circumference.

Answer 1

The Correct Option is B

We start with the given equation:
\(x^2 + y^2 - 18x - 16y + 120 = 0\)
Group \(x\)-terms and \(y\)-terms:
\((x^2 - 18x) + (y^2 - 16y) + 120 = 0\)
Complete the square for \(x\):
\(x^2 - 18x = x^2 - 18x + 81 - 81 = (x - 9)^2 - 81\)
Complete the square for \(y\):
\(y^2 - 16y = y^2 - 16y + 64 - 64 = (y - 8)^2 - 64\)
Substitute back:
\((x - 9)^2 - 81 + (y - 8)^2 - 64 + 120 = 0\)
\((x - 9)^2 + (y - 8)^2 - 25 = 0\)
\((x - 9)^2 + (y - 8)^2 = 25\)
This is the equation of a circle with radius \(r = \sqrt25 = 5\).
Circumference = \(2\pi r = 2\pi \times 5 = 10\pi\).