Question

a circle has the equation x² + y² + 4x - 2y - 20 = 0. (a) find the center (h,k) and radius r of the circle. (b) graph the circle. (c) find the intercepts, if any, of the graph. the radius of the circle is 5. (type an integer or a decimal.) (b) use the graphing tool to graph the circle. (c) what are the intercepts? select the correct choice and, if necessary, fill in the answer box to complete your choice. a. the intercept(s) is/are (type an ordered pair. use a comma to separate answers as needed. simplify your answers for each coordinate, using radicals as needed.) b. there is no intercept.

Explanation:

Step1: Rewrite the circle equation in standard form

The general equation of a circle is \(x^{2}+y^{2}+4x - 2y-20 = 0\). Complete the square for \(x\) and \(y\) terms.
For \(x\) - terms: \(x^{2}+4x=(x + 2)^{2}-4\).
For \(y\) - terms: \(y^{2}-2y=(y - 1)^{2}-1\).
So the equation becomes \((x + 2)^{2}-4+(y - 1)^{2}-1-20 = 0\), which simplifies to \((x + 2)^{2}+(y - 1)^{2}=25\).

Step2: Find 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.
Comparing \((x + 2)^{2}+(y - 1)^{2}=25\) with \((x - h)^{2}+(y - k)^{2}=r^{2}\), we have \(h=-2\), \(k = 1\), and \(r = 5\).
The center of the circle is \((-2,1)\) and the radius \(r = 5\).

Step3: Find the \(x\) - intercepts

Set \(y = 0\) in the equation \((x + 2)^{2}+(y - 1)^{2}=25\).
\((x + 2)^{2}+(0 - 1)^{2}=25\), so \((x + 2)^{2}+1 = 25\), then \((x + 2)^{2}=24\).
Taking the square root of both sides, \(x+2=\pm\sqrt{24}=\pm2\sqrt{6}\).
\(x=-2\pm2\sqrt{6}\). The \(x\) - intercepts are \((-2 + 2\sqrt{6},0)\) and \((-2-2\sqrt{6},0)\).

Step4: Find the \(y\) - intercepts

Set \(x = 0\) in the equation \((x + 2)^{2}+(y - 1)^{2}=25\).
\((0 + 2)^{2}+(y - 1)^{2}=25\), so \(4+(y - 1)^{2}=25\), then \((y - 1)^{2}=21\).
Taking the square root of both sides, \(y - 1=\pm\sqrt{21}\).
\(y=1\pm\sqrt{21}\). The \(y\) - intercepts are \((0,1+\sqrt{21})\) and \((0,1-\sqrt{21})\).

Answer:

(a) Center: \((-2,1)\), Radius: \(5\)
(b) To graph the circle, plot the center \((-2,1)\) and then use the radius of \(5\) to draw the circle.
(c) \(x\) - intercepts: \((-2 + 2\sqrt{6},0),(-2-2\sqrt{6},0)\); \(y\) - intercepts: \((0,1+\sqrt{21}),(0,1-\sqrt{21})\)