6. Plot the graph of the equation $x^{2}-y^{4}=\sqrt{18 x-x^{2}-81}$, that is, represent on the coordinate plane all points with coordinates $(x ; y)$ that satisfy this equation.
Answer. See the figure.

Solution. Transform the given equation by extracting a complete square under the root: $x^{2}-y^{4}=\sqrt{-(x-9)^{2}}$. The expression on the right side is meaningful only when $x=9$. Substituting this value into the equation, we get: $9^{2}-y^{4}=0$. Factor the left side: $(3-y)(3+y)\left(9+y^{2}\right)=0$. Hence, $y=3$ or $y=-3$. Therefore, the coordinates of only two points $(9 ; 3)$ or $(9 ;-3)$ satisfy the given equation. The graph of the equation is shown in the figure.

## Grading Criteria.

- Correct transformations and reasoning, and the graph is correctly constructed - 7 points.
- Correct transformations, but the value $y=-3$ is lost; only one point is indicated as the graph - 3 points.
- One or two suitable points are indicated, possibly with verification, but without other explanations or after incorrect transformations - 1 point.
- Correct transformations, but it is stated that the expression under the root (or on the right side after squaring) is negative and the graph is an empty set of points - 1 point.
- Reasoning leading to the indication of two points, but these points are somehow connected (e.g., by a segment) - 1 point.
- Two points are indicated without explanation and are somehow connected - 0 points.
- In all other cases - 0 points.