9.2. The graph of the function $y=x^{2}+a x+b$ is shown in the figure. It is known that the line $A B$ is perpendicular to the line $y=x$. Find the length of the segment $O C$.

---

The text has been translated while preserving the original formatting and line breaks.
Answer: 1.

Solution. Since $y(0)=b$, then $B(0 ; b)$. Now let's find the length of the segment $O A$.

First method. Since the line $A B$ is perpendicular to the line $y=x$, it is parallel to the line $y=-x$. Moreover, this line passes through the point $B(0 ; b)$. Therefore,

![](https://cdn.mathpix.com/cropped/2024_05_06_f571041068196027ec27g-1.jpg?height=403&width=397&top_left_y=907&top_left_x=1649)
it is described by the equation $y=-x+b$. Since $y=0$ when $x=b$, then $O A=-b$.

Second method. From the problem statement, it follows that the bisector of triangle $A O B$, drawn to the side $A B$, lies on the line $y=x$, and therefore coincides with the height of this triangle. Consequently, $O A=O B=-b$.

Thus, the number $b$ and the desired length $c$ of the segment $O C$ are the roots of the quadratic equation $x^{2}+a x+b=0$. By Vieta's theorem: $b c=b$. Since $b \neq 0$, then $c=1$.

## Grading criteria:  + correct answer and complete justified solution  $\pm$ correct answer and solution with minor gaps in justification  干 only proved that $O A=O B$, with no further progress  Ғ used the equality $O A=$ OB without proof, and then obtained the correct answer in a justified manner  干 the correct answer was obtained based on specific numerical values of a and b  - only the answer is provided