Let's solve the problem step-by-step.
### Step 1: Find the Radius of the Circle
The center of the circle is at [tex]\((8, 6)\)[/tex] and a point on the circle is [tex]\((12, 10)\)[/tex].
We can use the distance formula to find the radius:
[tex]\[ \text{Radius} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
where [tex]\((x_1, y_1) = (8,6)\)[/tex] and [tex]\((x_2, y_2) = (12, 10)\)[/tex].
### Step 2: Calculate the Coordinates of the Given Points
We need to determine whether each of the given points lies on the circle by comparing the distance from the center to the point with the radius.
### Step 3: Classify Each Given Point
#### (2, 4)
Calculate distance from center:
[tex]\[ \sqrt{(2 - 8)^2 + (4 - 6)^2} = \sqrt{36 + 4} = \sqrt{40} \approx 6.32 \][/tex]
This distance is not equal to the radius.
#### (10, 12)
Calculate distance from center:
[tex]\[ \sqrt{(10 - 8)^2 + (12 - 6)^2} = \sqrt{4 + 36} = \sqrt{40} \approx 6.32 \][/tex]
This distance is not equal to the radius.
#### (4, 10)
Calculate distance from center:
[tex]\[ \sqrt{(4 - 8)^2 + (10 - 6)^2} = \sqrt{16 + 16} = \sqrt{32} \approx 5.656854 \][/tex]
This distance is equal to the radius.
#### (12, 2)
Calculate distance from center:
[tex]\[ \sqrt{(12 - 8)^2 + (2 - 6)^2} = \sqrt{16 + 16} = \sqrt{32} \approx 5.656854 \][/tex]
This distance is equal to the radius.
### Conclusion
We can now correctly place each point in the table based on whether the distance from the center equals the radius.
[tex]\[
\begin{tabular}{|l|l|}
\hline
\text{point lies on the circle} & \text{point does not lie on the circle} \\
\hline
(4,10), (12,2) & (2,4), (10,12) \\
\hline
\end{tabular}
\][/tex]
