Question

which equation represents a circle with a center at (-4, 9) and a diameter of 10 units? (x - 9)^2 + (y + 4)^2 = 25 (x + 4)^2 + (y - 9)^2 = 25 (x - 9)^2 + (y + 4)^2 = 100 (x + 4)^2 + (y - 9)^2 = 100

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.

Step2: Identify the center and radius

Given the center \((-4,9)\), so \(h=-4\), \(k = 9\). Given diameter \(d = 10\), then radius \(r=\frac{d}{2}=\frac{10}{2}=5\), and \(r^2 = 25\).

Step3: Substitute into the standard form

Substitute \(h=-4\), \(k = 9\), \(r^2=25\) into \((x - h)^2+(y - k)^2=r^2\). We get \((x-(-4))^2+(y - 9)^2=25\), which simplifies to \((x + 4)^2+(y - 9)^2=25\).

Answer:

\((x + 4)^2+(y - 9)^2=25\) (the second option)