Task B-4.4. Determine the equation of the circle whose center is on the x-axis, the radius is $5 \sqrt{10}$, and which intercepts a chord of length 10 on the line $3 x-4 y+24=0$.
## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_6e5f8ba1616e2c4b87f4g-16.jpg?height=580&width=742&top_left_y=1968&top_left_x=667)

By the Pythagorean theorem in triangle $\triangle B P S$ we get $|P S|=15$.

(1 point)

The distance from the center of the circle to the line $3 x-4 y+24=0$ is 15. We write

$$
\left|\frac{3 x+24}{5}\right|=15
$$

Then we have

$$
\begin{aligned}
& 3 x+24=75 \text { or } 3 x+24=-75, \text { i.e., } \\
& x=17 \text { or } x=-33
\end{aligned}
$$

The desired circles are

$$
\begin{aligned}
& (x-17)^{2}+y^{2}=250 \\
& (x+33)^{2}+y^{2}=250
\end{aligned}
$$