## Task B-4.4.

The center of the circle $2 x^{2}+2 y^{2}-4 x+8 y-20=0$ is passed through by the lines given by the equations $m x-y+3=0$ and $x-n y+2=0$. Determine the angle between these lines.
## Solution.

The equation of the circle in the standard form is $(x-1)^{2}+(y+2)^{2}=15 \Rightarrow S(1,-2)$.

If we substitute the coordinates of the center into the equations of the lines

$$
y=m x+3 \text { and } y=\frac{1}{n} x+\frac{2}{n}
$$

we get $m=-5, n=-\frac{3}{2}$.

2 points

The equations of the given lines are

$p_{1} \ldots y=-5 x+3$ and $p_{2} \ldots y=-\frac{2}{3} x-\frac{4}{3}$.

1 point

The angle between the lines $p_{1}$ and $p_{2}$ is calculated from

$$
\operatorname{tg} \alpha=\left|\frac{k_{2}-k_{1}}{1+k_{1} k_{2}}\right|=1
$$

The desired angle is $45^{\circ}$. 1 point