## Task B-4.1.

The endpoints of a chord of the circle $k$ have coordinates $(6,0)$ and $(6,10)$. Determine the equation of the circle $k$ if the distance from its center to the chord is 4.
## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_4b7629e40479ae2c2357g-25.jpg?height=816&width=1216&top_left_y=1017&top_left_x=340)

The sketch clearly shows a right triangle with legs of lengths 4 and 5, and the conclusion that \( q \) is obviously equal to 5.

Applying the Pythagorean theorem to the right triangle \( B S C \), we get \( r = \sqrt{25 + 16} = \sqrt{41} \).

Since point \( C \) clearly has coordinates \((6, 5)\) and the distance from the center of the circle to the point is 4, we conclude that \( p = 6 - 4 = 2 \) or \( p = 6 + 4 = 10 \).

Thus, we have two possible solutions:

\[
\begin{aligned}
& k_{1} \ldots (x-2)^{2} + (y-5)^{2} = 41 \text{ and} \\
& k_{1} \ldots (x-10)^{2} + (y-5)^{2} = 41
\end{aligned}
\]