Question

given a circle with center (6, 2) and radius $sqrt{19}$, (a) write an equation of the circle in standard form. (b) graph the circle.

Explanation:

Step1: Recall the standard - form equation of a circle

The standard - form equation of a circle is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center of the circle and $r$ is the radius.

Step2: Identify the values of $h$, $k$, and $r$

Given that the center is $(6,2)$ and the radius $r=\sqrt{19}$, we have $h = 6$, $k = 2$, and $r^2=19$.

Step3: Substitute the values into the standard - form equation

Substituting $h = 6$, $k = 2$, and $r^2 = 19$ into $(x - h)^2+(y - k)^2=r^2$, we get $(x - 6)^2+(y - 2)^2=19$. For part (b), the key is to use the center and radius to plot points and draw the circle.

Answer:

(a) $(x - 6)^2+(y - 2)^2=19$
(b) To graph the circle:

1. Locate the center of the circle at the point $(6,2)$ on the coordinate - plane.
2. From the center, use the radius $r = \sqrt{19}\approx4.36$ to find points on the circle. Move approximately $4.36$ units up, down, left, and right from the center point $(6,2)$ and sketch the circle passing through these points.