## Task B-4.3.

The center of the circle passing through points $A(0,5)$ and $B(0,-1)$ lies on the line $y=2x-6$.

Determine the tangent of the smaller arc angle over the chord $\overline{A B}$.
## First solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_9b9f9f32e16538559f3ag-23.jpg?height=471&width=743&top_left_y=2055&top_left_x=565)

The line $y=2$ is perpendicular to the chord $\overline{A B}$ and passes through the center of the given circle, so the ordinate of the center is $q=2$.

The coordinates of the center $(p, q)$ satisfy the equation of the line $y=2x-6$, so we have $q=2p-6$,

$2=2p-6$ and

$p=4$.

1 point

The smaller inscribed angle over the chord $\overline{A B}$ is equal to half of the central angle $A S B$ over that chord, or equal to the angle $\alpha=\varangle P S A$.

Finally, $\operatorname{tg} \alpha=\frac{3}{4}$.