Question

a circle has the equation x² + y² = 16. (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. (a) the center of the circle is . (type an ordered pair, using integers or decimals.)

Explanation:

Step1: Recall the standard form of a circle equation

The standard - form of a circle equation is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center of the circle and $r$ is the radius. The given equation is $x^{2}+y^{2}=16$, which can be written as $(x - 0)^2+(y - 0)^2 = 4^2$.

Step2: Identify the center and radius

Comparing with the standard form, we have $h = 0$, $k = 0$, and $r=4$.

Step3: Find the x - intercepts

Set $y = 0$ in the equation $x^{2}+y^{2}=16$. Then $x^{2}+0^{2}=16$, so $x^{2}=16$, and $x=\pm4$. The x - intercepts are $(-4,0)$ and $(4,0)$.

Step4: Find the y - intercepts

Set $x = 0$ in the equation $x^{2}+y^{2}=16$. Then $0^{2}+y^{2}=16$, so $y^{2}=16$, and $y=\pm4$. The y - intercepts are $(0, - 4)$ and $(0,4)$.

Answer:

(a) $(0,0)$
(b) To graph the circle, plot the center at the origin $(0,0)$ and then use the radius $r = 4$ to draw all the points that are 4 units away from the center.
(c) x - intercepts: $(-4,0)$ and $(4,0)$; y - intercepts: $(0, - 4)$ and $(0,4)$