Question

a circle has the equation x² + y² = 121. (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 - equation form

The standard form of a circle's equation is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center and $r$ is the radius. The given equation is $x^{2}+y^{2}=121$, which can be written as $(x - 0)^2+(y - 0)^2 = 11^2$.

Step2: Identify the center

Comparing with the standard - form, when $h = 0$ and $k = 0$, the center of the circle $(h,k)=(0,0)$.

Step3: Identify the radius

Since $r^{2}=121$, taking the square root of both sides (and considering the positive value for the radius), we have $r=\sqrt{121}=11$.

Step4: Find the x - intercepts

Set $y = 0$ in the equation $x^{2}+y^{2}=121$. Then $x^{2}+0^{2}=121$, so $x^{2}=121$. Solving for $x$, we get $x=\pm11$. The x - intercepts are $(-11,0)$ and $(11,0)$.

Step5: Find the y - intercepts

Set $x = 0$ in the equation $x^{2}+y^{2}=121$. Then $0^{2}+y^{2}=121$, so $y^{2}=121$. Solving for $y$, we get $y=\pm11$. The y - intercepts are $(0, - 11)$ and $(0,11)$.

Answer:

(a) $(0,0)$
(b) To graph the circle: Plot the center at the origin $(0,0)$. Then, using a compass with a radius of $11$ units, draw a circle around the center.
(c) x - intercepts: $(-11,0)$ and $(11,0)$; y - intercepts: $(0, - 11)$ and $(0,11)$