## Task 2 - 240932

In a Cartesian coordinate system, let the circle $k$ be centered at the origin with a radius of $\sqrt{2}$, and the line $g$ be given by the equation $y = -x + 10$.

Determine the equations for the two tangents to $k$ that are parallel to $g$!
The lines parallel to $g$ have the same slope as $g$, which is -1. Furthermore, the radii at the points of tangency are perpendicular to the tangents, thus having a slope of 1 and passing through the center of the circle, and therefore have the equation $y_{r}=x$, which means the points of tangency are located at the coordinates (1,1) and (-1,-1). (It is easy to verify that these points are $\sqrt{2}$ units away from the center of the circle, which is the origin, and thus lie on $k$.)

Thus, the two tangents have the equations

$$
y_{t_{1}}=-(x-1)+1=-x+2 \quad \text { and } \quad y_{t_{2}}=-(x+1)-1=-x-2
$$