Question

which equation represents a circle with center t(5,-1) and a radius of 16 units?
a. $(x - 5)^2 + (y + 1)^2 = 16$
b. $(x - 5)^2 + (y + 1)^2 = 256$
c. $(x + 5)^2 + (y - 1)^2 = 16$
d. $(x + 5)^2 + (y - 1)^2 = 256$

Explanation:

Step1: Recall the circle equation formula

The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\).

Step2: Identify \(h\), \(k\), and \(r\)

Given center \(T(5, -1)\), so \(h = 5\), \(k = -1\), and radius \(r = 16\).

Step3: Substitute into the formula

Substitute \(h = 5\), \(k = -1\), and \(r = 16\) into the formula. First, calculate \(r^2\): \(r^2 = 16^2 = 256\). Then, substitute \(h\), \(k\), and \(r^2\) into \((x - h)^2 + (y - k)^2 = r^2\), we get \((x - 5)^2 + (y - (-1))^2 = 256\), which simplifies to \((x - 5)^2 + (y + 1)^2 = 256\).

Answer:

B. \((x - 5)^2 + (y + 1)^2 = 256\)