Question

a circle has the equation x² + y² = 25. (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 circle - standard form

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. The given equation $x^{2}+y^{2}=25$ can be written as $(x - 0)^2+(y - 0)^2 = 5^2$.

Step2: Identify center and radius

Comparing with the standard - form, we have $h = 0$, $k = 0$, and $r = 5$.

Step3: Find x - intercepts

Set $y = 0$ in the equation $x^{2}+y^{2}=25$. Then $x^{2}+0^{2}=25$, so $x^{2}=25$, and $x=\pm5$. The x - intercepts are $(-5,0)$ and $(5,0)$.

Step4: Find y - intercepts

Set $x = 0$ in the equation $x^{2}+y^{2}=25$. Then $0^{2}+y^{2}=25$, so $y^{2}=25$, and $y=\pm5$. The y - intercepts are $(0, - 5)$ and $(0,5)$.

Answer:

(a) $(0,0)$
(b) To graph the circle, plot the center at the origin $(0,0)$ and then use the radius $r = 5$ to draw a circle around the center. Mark the x - intercepts at $x=-5$ and $x = 5$ and the y - intercepts at $y=-5$ and $y = 5$.
(c) x - intercepts: $(-5,0),(5,0)$; y - intercepts: $(0, - 5),(0,5)$