Question

for the equation x² + y² - 4x - 8y - 16 = 0, do the following. (a) find the center (h,k) and radius r of the circle. (b) graph the circle. (c) find the intercepts, if any.

Explanation:

Step1: Rewrite the equation in standard form

Complete the square for x and y terms.

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Step2: Identify the center and radius

The standard - form of a circle is \((x - h)^{2}+(y - k)^{2}=r^{2}\), where \((h,k)\) is the center and \(r\) is the radius.
For \((x - 2)^{2}+(y - 4)^{2}=36\), we have \(h = 2\), \(k = 4\) and \(r=\sqrt{36}=6\).

Step3: Find the x - intercepts

Set \(y = 0\) in the original equation \((x - 2)^{2}+(0 - 4)^{2}=36\).

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Step4: Find the y - intercepts

Set \(x = 0\) in the original equation \((0 - 2)^{2}+(y - 4)^{2}=36\).

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Answer:

(a) Center \((h,k)=(2,4)\), radius \(r = 6\)
(b) To graph the circle, plot the center \((2,4)\) and then draw a circle with radius \(r = 6\) around it.
(c) x - intercepts: \(x=2 + 2\sqrt{5}\) and \(x=2-2\sqrt{5}\); y - intercepts: \(y=4 + 4\sqrt{2}\) and \(y=4-4\sqrt{2}\)