10.1. The graph of the quadratic function $y=a x^{2}+c$ intersects the coordinate axes at the vertices of an equilateral triangle. What is the value of ac?
Answer: -3.

Solution. Since the graph intersects the OX axis at two points, the numbers a and c have different signs. The points of intersection are: $A\left(\sqrt{-\frac{c}{a}} ; 0\right) ; B\left(-\sqrt{-\frac{c}{a}} ; 0\right)$. The graph intersects the OY axis at point $C(0 ; c)$. Then the side $AB$ of the equilateral triangle is $2 \sqrt{-\frac{c}{a}}$, and its height is $|c|$. Since the side $a$ and the height $h$ of an equilateral triangle are related by $h=\frac{a \sqrt{3}}{2}$, we have $|c|=\frac{\sqrt{3}}{2} \cdot 2 \sqrt{-\frac{c}{a}}$, from which $c^{2}=-3 \cdot \frac{c}{a}$, so $a c=-3$.

There are other ways to obtain the equation relating a and c, for example, by using that $A B=A C$.

Grading criteria.

“+” A complete and well-reasoned solution is provided

“士” A generally correct reasoning is provided, containing minor gaps or inaccuracies (e.g., only considering $s > 0$)

“Ғ” Correct reasoning, but a computational error is made

“ๆ” Only the correct answer is provided

“-” The problem is not solved or is solved incorrectly