Question

a circle is represented by the equation below: (x + 8)² + (y - 3)² = 100 which statement is true? the circle is centered at (-8, 3) and has a diameter of 20. the circle is centered at (8, -3) and has a radius of 20. the circle is centered at (8, -3) and has a diameter of 20. the circle is centered at (-8, 3) and has a radius of 20.

Explanation:

Step1: Recall the standard form of a circle's 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.
For the given equation \((x + 8)^2+(y - 3)^2=100\), we can rewrite it as \((x-(-8))^2+(y - 3)^2 = 10^2\).

Step2: Identify the center and radius

By comparing with the standard form \((x - h)^2+(y - k)^2=r^2\), we get \(h=-8\), \(k = 3\), and \(r = 10\) (since \(r^2=100\) implies \(r=\sqrt{100}=10\)). The diameter \(d = 2r\), so \(d=2\times10 = 20\).

Answer:

The circle is centered at \((-8,3)\) and has a diameter of \(20\).