Factorization
[ [ Rafiki and GMT on the coordinate plane]

Find the minimum value of $x^{2}+y^{2}$, if $x^{2}-y^{2}+6 x+4 y+5=0$.

#
$x^{2}-y^{2}+6 x+4 y+5=(x+3)^{2}-(y-2)^{2}=(x+y+1)(x-y+5)=0$.

Thus, the graph of the obtained equation consists of two lines $y=-x-1$ and $y=x+5$, which intersect the y-axis at the points

$(0,-1)$ and $(0,5)$ (see figure). $x^{2}+y^{2}$ is the square of the distance from the point $M(x, y)$ to the origin, so its value will be the smallest when $M$ is the foot of the perpendicular dropped from the point $O(0,0)$ to the nearest line to this point. Considering that both lines cut off isosceles right triangles from the coordinate axes with legs of 1 and 5, we get that the line $y=-x-1$ is closer to the point $O$, then $O M=0.5$.

![](https://cdn.mathpix.com/cropped/2024_05_06_8c2fdb390ca80df56784g-13.jpg?height=465&width=581&top_left_y=545&top_left_x=747)

## Answer

0.5 .