Question

a circle has the equation x² + y²+4x - 2y - 4 = 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 3. (type an ordered - pair, using integers or decimals.) (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 equation in standard form

The general equation of a circle is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center and $r$ is the radius. Given $x^{2}+y^{2}+4x - 2y-4 = 0$. Complete the square for $x$ and $y$ terms.
For the $x$ - terms: $x^{2}+4x=(x + 2)^{2}-4$. For the $y$ - terms: $y^{2}-2y=(y - 1)^{2}-1$.
So the equation becomes $(x + 2)^{2}-4+(y - 1)^{2}-1-4 = 0$, which simplifies to $(x + 2)^{2}+(y - 1)^{2}=9$.

Step2: Find the center and radius

Comparing $(x + 2)^{2}+(y - 1)^{2}=9$ with $(x - h)^2+(y - k)^2=r^2$, we have $h=-2,k = 1,r = 3$. The center $(h,k)$ is $(-2,1)$ and the radius $r = 3$.

Step3: Find the $x$ - intercepts

Set $y = 0$ in the equation $(x + 2)^{2}+(y - 1)^{2}=9$. Then $(x + 2)^{2}+(0 - 1)^{2}=9$, so $(x + 2)^{2}+1=9$, $(x + 2)^{2}=8$, $x+2=\pm\sqrt{8}=\pm2\sqrt{2}$, and $x=-2\pm2\sqrt{2}$. The $x$ - intercepts are $(-2 + 2\sqrt{2},0)$ and $(-2-2\sqrt{2},0)$.

Step4: Find the $y$ - intercepts

Set $x = 0$ in the equation $(x + 2)^{2}+(y - 1)^{2}=9$. Then $(0 + 2)^{2}+(y - 1)^{2}=9$, $4+(y - 1)^{2}=9$, $(y - 1)^{2}=5$, $y-1=\pm\sqrt{5}$, and $y=1\pm\sqrt{5}$. The $y$ - intercepts are $(0,1+\sqrt{5})$ and $(0,1 - \sqrt{5})$.

Answer:

(a) Center: $(-2,1)$, Radius: $3$
(b) Graph the circle with center $(-2,1)$ and radius $3$ (using a graphing tool as instructed in the problem - this step is more visual and not shown in full here but can be done on a graphing calculator or software)
(c) $x$ - intercepts: $(-2 + 2\sqrt{2},0),(-2-2\sqrt{2},0)$; $y$ - intercepts: $(0,1+\sqrt{5}),(0,1 - \sqrt{5})$