Question

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

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

Step3: Substitute into the standard form

Substitute \(h=-4\), \(k = 9\), \(r = 5\) into \((x - h)^2+(y - k)^2=r^2\).
We get \((x-(-4))^2+(y - 9)^2=5^2\), which simplifies to \((x + 4)^2+(y - 9)^2=25\).

Answer:

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