1. The problem is to understand the equation and properties of a hyperbolic paraboloid. 2. A hyperbolic paraboloid is a type of saddle surface that can be described by the equation $$z=\frac{x^2}{a^2}-\frac{y^2}{b^2}$$ where $a$ and $b$ are constants that determine the curvature along the $x$ and $y$ axes. 3. Important rules: - The surface curves upward in one direction and downward in the perpendicular direction. - It is a doubly ruled surface, meaning it can be generated by two families of straight lines. 4. To analyze the shape, consider cross sections: - For fixed $y$, the cross section is a parabola opening upward: $$z=\frac{x^2}{a^2}-\frac{y^2}{b^2} \Rightarrow z=\frac{x^2}{a^2} - \text{constant}$$ - For fixed $x$, the cross section is a parabola opening downward: $$z=\text{constant} - \frac{y^2}{b^2}$$ 5. This explains the saddle shape: one direction curves up, the other curves down. 6. Example: If $a=b=1$, the equation simplifies to $$z=x^2 - y^2$$ which is the standard hyperbolic paraboloid. Final answer: The hyperbolic paraboloid is described by $$z=\frac{x^2}{a^2}-\frac{y^2}{b^2}$$ and has a saddle shape with parabolic cross sections in perpendicular directions.