Question

a circle in the xy-plane has a diameter with endpoints (2,4) and (2,14). an equation of this circle is ((x - 2)^2 + (y - 9)^2 = r^2), where (r) is a positive constant. what is the value of (r)?

Explanation:

Step1: Find the length of the diameter

The endpoints of the diameter are \((2, 4)\) and \((2, 14)\). Since the \(x\)-coordinates are the same, the distance between them (length of the diameter) is the difference in \(y\)-coordinates. So, \(d = 14 - 4 = 10\).

Step2: Find the radius

The radius \(r\) is half of the diameter. So, \(r=\frac{d}{2}=\frac{10}{2} = 5\).

Answer:

\(5\)