B1. In the equation of the line $x+m y-4=0$, determine $m$ so that the distance between the intersection points of the line $\mathrm{s}$ with the coordinate axes is equal to $2 \sqrt{5}$.
B1. It is clear that $m \neq 0$, otherwise the line would not intersect the y-axis. Under this condition, the line intersects the y-axis $\mathrm{v}$ at the point $N\left(0, \frac{4}{m}\right)$. It intersects the x-axis at the point $M(4,0)$. The equation $\sqrt{16+\frac{16}{m^{2}}}=2 \sqrt{5}$ must hold, from which we can express $m^{2}=4$ or $m= \pm 2$.

![](https://cdn.mathpix.com/cropped/2024_06_07_ce887fced8b2a2bf0894g-12.jpg?height=388&width=505&top_left_y=314&top_left_x=1415)

Intersections with the coordinate axes: $N\left(0, \frac{4}{m}\right)$ and $M(4,0)$

1+1 point

Points, substituted into the formula $d(M, N)=\sqrt{16+\frac{16}{m^{2}}}=\sqrt{\frac{16 m^{2}+16}{m^{2}}}$ 1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_ce887fced8b2a2bf0894g-12.jpg?height=80&width=1639&top_left_y=859&top_left_x=274)

![](https://cdn.mathpix.com/cropped/2024_06_07_ce887fced8b2a2bf0894g-12.jpg?height=66&width=1639&top_left_y=935&top_left_x=274)

![](https://cdn.mathpix.com/cropped/2024_06_07_ce887fced8b2a2bf0894g-12.jpg?height=45&width=1639&top_left_y=997&top_left_x=274)

NOTE: If both solutions are not written, the last point will not be awarded.