Answer
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Work Step by Step
Quadric surfaces are a class of three-dimensional geometric shapes defined by second-degree polynomial equations in three variables. They are called "quadric" because the highest power of any variable in the equation is two.
1. Ellipsoids:
- Equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$
- Sketch: An ellipsoid is a three-dimensional analogue of an ellipse, resembling a stretched or squashed sphere. It has three principal axes of different lengths.
2. Hyperboloids of one sheet:
- Equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$
- Sketch: A hyperboloid of one sheet consists of two connected, curved surfaces that resemble a saddle shape. It opens up or down along the z-axis.
3. Hyperboloids of two sheets:
- Equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1$
- Sketch: Similar to the hyperboloid of one sheet, the hyperboloid of two sheets consists of two connected, curved surfaces. However, it opens up or down along the x and y axes.
4. Elliptic paraboloids:
- Equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = z$
- Sketch: An elliptic paraboloid is shaped like a bowl or a cup. It opens upwards or downwards along the z-axis and has elliptical cross-sections.
5. Hyperbolic paraboloids:
- Equation: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = z$
- Sketch: A hyperbolic paraboloid resembles two intersecting straight lines or a saddle shape. It opens upwards along the x-axis and downwards along the y-axis.
6. Cones:
- Equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 0$
- Sketch: A cone is a three-dimensional shape with a circular base that narrows to a point called the apex. It can be upright or inverted, depending on the signs of the coefficients.
These examples represent different types of quadric surfaces, each with its own distinct shape and equation.
