## Task A-2.3.

The cheese is located at the point $(13,13)$, and the mouse runs straight from the point $(4,-2)$ to the point $(-1,23)$. At which point is the mouse closest to the cheese?
## First Solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_21f39afeb1dfc245d841g-08.jpg?height=705&width=594&top_left_y=1298&top_left_x=660)

The equation of the line along which the mouse runs is determined by the points $(4,-2)$ and $(-1,23)$:

$$
y-(-2)=\frac{23-(-2)}{-1-4}(x-4)
$$

from which, after simplification, we get $y=-5x+18$.

We need to determine a real number $x$ such that the point $(x, -5x + 18)$ lies on the segment defined by the points $(4, -2)$ and $(-1, 23)$, and the distance between the points $(13, 13)$ and $(x, -5x + 18)$ is minimized. The distance between these points is

$$
\sqrt{(x-13)^2 + (-5x + 18 - 13)^2} = \sqrt{26x^2 - 76x + 194}
$$

The minimum of the quadratic function (under the square root) is achieved at its vertex.

For the quadratic function $26x^2 - 76x + 194$, the minimum is achieved at the point $x_0 = \frac{19}{13}$.

For the point $\left(x_0, -5x_0 + 18\right)$ to lie on the segment, it must hold that $-1 \leq x_0 \leq 4$. Since indeed $-1 \leq \frac{19}{13} \leq 4$, we conclude that this is our desired $x_0$.

Finally, the desired point is $\left(\frac{19}{13}, \frac{139}{13}\right)$.

1 point

Note: The problem can also be interpreted as follows: the goal is to find the radius $r$ of a circle with center at $(13, 13)$ such that this circle is tangent to the segment defined by the points $(4, -2)$ and $(-1, 23)$. The condition for tangency of the circle and the line is obtained by requiring that the discriminant of the resulting quadratic equation is zero (otherwise, the circle intersects the line twice or not at all, depending on the sign of the discriminant).

This is precisely how the formula for calculating the distance $r$ from a point $S(p, q)$ to a line $y = kx + l$ is derived:

$$
r = \frac{|q - kp - l|}{\sqrt{1 + k^2}}
$$

The scoring scheme for this solution is similar to the above: 1 point for finding the equation of the line, 1 point for stating or deriving the formula for the distance from a point to a line, 1 point for determining this distance, 1 point for determining the coordinate $x_0$, 1 point for justifying that this point lies on the segment (and not just on the line), and 1 point for determining the ordinate of the desired point.