Question

a circle has the equation x² + y²+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. (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. type exact answers for each coordinate, using radicals as needed. simplify your answer.) b. there is no intercept

Explanation:

Step1: Rewrite the circle equation in standard form

The general equation of a circle is \(x^{2}+y^{2}+2x - 8y-19 = 0\). Complete the square for \(x\) and \(y\) terms.
For the \(x\) - terms: \(x^{2}+2x=(x + 1)^{2}-1\).
For the \(y\) - terms: \(y^{2}-8y=(y - 4)^{2}-16\).
So the equation becomes \((x + 1)^{2}-1+(y - 4)^{2}-16-19 = 0\), which simplifies to \((x + 1)^{2}+(y - 4)^{2}=36\).

Step2: Find the center and radius

The standard - form of a circle equation is \((x - h)^{2}+(y - k)^{2}=r^{2}\), where \((h,k)\) is the center and \(r\) is the radius.
Comparing \((x + 1)^{2}+(y - 4)^{2}=36\) with \((x - h)^{2}+(y - k)^{2}=r^{2}\), we have \(h=-1\), \(k = 4\), and \(r = 6\).
So the center \((h,k)=(-1,4)\) and radius \(r = 6\).

Step3: Find the \(x\) - intercepts

Set \(y = 0\) in the equation \((x + 1)^{2}+(y - 4)^{2}=36\).
We get \((x + 1)^{2}+(0 - 4)^{2}=36\), i.e., \((x + 1)^{2}+16 = 36\), then \((x + 1)^{2}=20\).
Taking the square root of both sides, \(x+1=\pm\sqrt{20}=\pm2\sqrt{5}\).
So \(x=-1\pm2\sqrt{5}\). 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 \((x + 1)^{2}+(y - 4)^{2}=36\).
We have \((0 + 1)^{2}+(y - 4)^{2}=36\), i.e., \(1+(y - 4)^{2}=36\), then \((y - 4)^{2}=35\).
Taking the square root of both sides, \(y - 4=\pm\sqrt{35}\).
So \(y=4\pm\sqrt{35}\). The \(y\) - intercepts are \((0,4+\sqrt{35})\) and \((0,4-\sqrt{35})\).

Answer:

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