To determine the length of the radius of a circle with its center at [tex]\( 2 + 3i \)[/tex] and a point on the circle at [tex]\( 7 + 2i \)[/tex], we need to find the distance between these two points in the complex plane. This distance corresponds to the radius of the circle.
The distance [tex]\( d \)[/tex] between two points [tex]\( a = x_1 + y_1i \)[/tex] and [tex]\( b = x_2 + y_2i \)[/tex] in the complex plane can be found using the distance formula:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
For our two points:
- The center of the circle is [tex]\( 2 + 3i \)[/tex] where [tex]\( x_1 = 2 \)[/tex] and [tex]\( y_1 = 3 \)[/tex].
- The point on the circle is [tex]\( 7 + 2i \)[/tex] where [tex]\( x_2 = 7 \)[/tex] and [tex]\( y_2 = 2 \)[/tex].
Now substitute [tex]\( x_1 \)[/tex], [tex]\( y_1 \)[/tex], [tex]\( x_2 \)[/tex], and [tex]\( y_2 \)[/tex] into the distance formula:
[tex]\[
d = \sqrt{(7 - 2)^2 + (2 - 3)^2}
\][/tex]
Simplifying inside the square root:
[tex]\[
d = \sqrt{(5)^2 + (-1)^2}
\][/tex]
[tex]\[
d = \sqrt{25 + 1}
\][/tex]
[tex]\[
d = \sqrt{26}
\][/tex]
Therefore, the length of the radius of the circle is [tex]\( \sqrt{26} \)[/tex]. This corresponds to the second option provided in the question:
[tex]\[ \boxed{\sqrt{26}} \][/tex]
