Question

a circle has the equation (x^{2}+y^{2}+2x - 8y-19 = 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. (d) what are the intercepts? select the correct choice and, if necessary, fill in the answer box to complete your choice. a. the (x)-intercept(s) is/are (type an ordered pair. use a comma to separate answers as needed. type exact answers for each coordinate, using radicals as needed.) b. there is no (x)-intercept. the radius of the circle is 6. (type an integer or a decimal.) you are to use the graphing tool to graph the circle.

Explanation:

Step1: Rewrite the circle equation in standard form

The general equation of a circle is \(x^{2}+y^{2}+Dx + Ey+F = 0\), and its standard - form is \((x - a)^{2}+(y - b)^{2}=r^{2}\), where \((a,b)\) is the center and \(r\) is the radius. Given \(x^{2}+y^{2}+2x - 8y-19 = 0\). Completing the square for \(x\) and \(y\) terms:
\[

$$\begin{align*} x^{2}+2x+y^{2}-8y&=19\\ x^{2}+2x + 1+y^{2}-8y+16&=19 + 1+16\\ (x + 1)^{2}+(y - 4)^{2}&=36 \end{align*}$$

\]

Step2: Find the center and radius

Comparing \((x + 1)^{2}+(y - 4)^{2}=36\) with \((x - a)^{2}+(y - b)^{2}=r^{2}\), we have \(a=-1\), \(b = 4\) and \(r^{2}=36\), so \(r = 6\).

Center \((h,k)=(-1,4)\), radius \(r = 6\)

Step3: Find the \(x\) - intercepts

Set \(y = 0\) in the equation \((x + 1)^{2}+(y - 4)^{2}=36\). Then \((x + 1)^{2}+(0 - 4)^{2}=36\), \((x + 1)^{2}+16=36\), \((x + 1)^{2}=20\), \(x+1=\pm\sqrt{20}=\pm2\sqrt{5}\), \(x=-1\pm2\sqrt{5}\). The \(x\) - intercepts are \((-1 + 2\sqrt{5},0)\) and \((-1-2\sqrt{5},0)\)

Step4: Find the \(y\) - intercepts

Set \(x = 0\) in the equation \((x + 1)^{2}+(y - 4)^{2}=36\). Then \((0 + 1)^{2}+(y - 4)^{2}=36\), \(1+(y - 4)^{2}=36\), \((y - 4)^{2}=35\), \(y-4=\pm\sqrt{35}\), \(y=4\pm\sqrt{35}\). The \(y\) - intercepts are \((0,4+\sqrt{35})\) and \((0,4 - \sqrt{35})\)

Answer:

(a) Center \((-1,4)\), radius \(r = 6\)
(b) To graph the circle, plot the center \((-1,4)\) and then use the radius of \(6\) units to draw the circle around the center.
(c) \(x\) - intercepts: \((-1 + 2\sqrt{5},0),(-1-2\sqrt{5},0)\); \(y\) - intercepts: \((0,4+\sqrt{35}),(0,4 - \sqrt{35})\)