To solve for the value of [tex]\( r \)[/tex] in the given circle equation, we need to find the radius of the circle.
Let's start by identifying the coordinates of the endpoints of the diameter: [tex]\((2, 4)\)[/tex] and [tex]\((2, 14)\)[/tex].
1. Find the distance between the two endpoints of the diameter:
The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the plane is:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Plugging in our points:
[tex]\[
d = \sqrt{(2 - 2)^2 + (14 - 4)^2}
\][/tex]
Since the [tex]\( x \)[/tex]-coordinates are the same, the formula simplifies to:
[tex]\[
d = \sqrt{0 + (14 - 4)^2} = \sqrt{10^2} = 10
\][/tex]
Thus, the diameter of the circle is [tex]\( 10 \)[/tex].
2. Calculate the radius of the circle:
The radius [tex]\( r \)[/tex] is half of the diameter. So,
[tex]\[
r = \frac{\text{diameter}}{2} = \frac{10}{2} = 5
\][/tex]
Therefore, the radius [tex]\( r \)[/tex] of the circle is [tex]\( 5 \)[/tex].
So, the value of [tex]\( r \)[/tex] is [tex]\( \boxed{5} \)[/tex].
