Question

what is the length of the radius of a circle with a center at 2 + 3i and a point on the circle at 7 + 2i?
2√6
√26
√56
10

Explanation:

Step1: Recall distance formula

The distance $d$ between two complex - numbers $z_1=a + bi$ and $z_2=c+di$ is $d=\sqrt{(c - a)^2+(d - b)^2}$. Here, the center of the circle $z_1 = 2+3i$ (so $a = 2$, $b = 3$) and a point on the circle $z_2=7 + 2i$ (so $c = 7$, $d = 2$).

Step2: Substitute values

Substitute $a = 2$, $b = 3$, $c = 7$, $d = 2$ into the formula: $d=\sqrt{(7 - 2)^2+(2 - 3)^2}$.

Step3: Calculate the squares

First, calculate $(7 - 2)^2=5^2 = 25$ and $(2 - 3)^2=(-1)^2 = 1$.

Step4: Find the sum and square - root

Then $d=\sqrt{25 + 1}=\sqrt{26}$. Since the radius of a circle is the distance between the center and a point on the circle, the radius is $\sqrt{26}$.

Answer:

B. $\sqrt{26}$