Question

example #3 determine the diameter of the circle with the equation \\((x - 3)^2 + (y - 5)^2 = 49\\). how did you find the diameter of the circle given the equation?

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.
Given equation: \((x - 3)^2 + (y - 5)^2 = 49\).

Step2: Identify the radius

Compare with the standard form. Here, \(r^2 = 49\), so take the square root: \(r = \sqrt{49} = 7\).

Step3: Calculate the diameter

Diameter \(d\) is twice the radius: \(d = 2r\). Substitute \(r = 7\): \(d = 2\times7 = 14\).

Answer:

The diameter of the circle is 14. To find it, first identify the radius from the circle's equation \((x - 3)^2 + (y - 5)^2 = 49\) (since \(r^2 = 49\), \(r = 7\)), then multiply the radius by 2 (\(d = 2\times7 = 14\)).