Question

a circle with center r is graphed in the xy - plane. which of the following is an equation of the circle? choose 1 answer: (a) (x + 3)^2+(y - 4)^2 = 5 (b) (x - 3)^2+(y + 4)^2 = 5 (c) (x + 3)^2+(y - 4)^2 = 25 (d) (x - 3)^2+(y + 4)^2 = 25

Explanation:

Step1: Identify center - point

From the graph, the center of the circle $R$ has coordinates $(3,-4)$. 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. Here $h = 3$ and $k=-4$, so the left - hand side of the equation is $(x - 3)^2+(y+4)^2$.

Step2: Find the radius

The circle passes through the point $(0, - 8)$. The distance between the center $(3,-4)$ and the point $(0,-8)$ is the radius $r$. Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, we have $r=\sqrt{(3 - 0)^2+(-4+8)^2}=\sqrt{9 + 16}=\sqrt{25}=5$. Then $r^2 = 25$.

Answer:

D. $(x - 3)^2+(y + 4)^2=25$