Question

1. which of the following is the equation in standard form of a circle with center (5,-2) and radius equal to 8 ? a. $(x - 5)^2 + (y + 2)^2 = 8$ b. $\frac{(x + 5)^2}{8} + \frac{(y - 2)^2}{8} = 2$ c. $(x - 5)^2 + (y - 2)^2 = 64$ d. $sqrt{(x - 5)^2 + (y + 2)^2} = 16$ e. $(5x + 2)^2 + (2y + 5)^2 = 128$ f. $(x - 5)^2 + (y + 2)^2 = 64$ g. $(x + 2)^2 + (y - 5)^2 = 64$

Explanation:

Step1: Recall the standard form of a circle's equation

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 the values of \(h\), \(k\), and \(r\)

Given the center \((5, -2)\), so \(h = 5\) and \(k = -2\). The radius \(r = 8\), so \(r^2 = 8^2 = 64\).

Step3: Substitute the values into the standard form equation

Substituting \(h = 5\), \(k = -2\), and \(r^2 = 64\) into \((x - h)^2 + (y - k)^2 = r^2\), we get \((x - 5)^2 + (y - (-2))^2 = 64\), which simplifies to \((x - 5)^2 + (y + 2)^2 = 64\).

Answer:

F. \((x - 5)^2 + (y + 2)^2 = 64\)