Answer
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Step 1:I'll solve this problem by providing the equations and explanations for each surface as shown in the image.
Step 2:: Plane
* Each point $$(x,y,z)$$ satisfying the equation lies on the plane
- Coordinate System: Cartesian coordinates - Explanation: This linear equation represents a flat surface where:
Step 3:: Sphere
* All points $$(x,y,z)$$ are equidistant from the center
- Coordinate System: Spherical coordinates - Explanation: This equation describes a sphere:
Step 4:: Elliptic Paraboloid
* $$a$$ and $$b$$ control the curvature in $$x$$ and $$y$$ directions
- Coordinate System: Cartesian coordinates - Explanation: This surface has a parabolic shape: * The surface opens upward or downward depending on the sign of the coefficients
Step 5:: Hyperbolic Paraboloid
- Equation: $$z = \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}
- Coordinate System: Cartesian coordinates - Explanation: This surface has a saddle-like shape: * Similar to elliptic paraboloid, but with opposite signs * Creates a distinctive "saddle" or "horse-back" shape
Step 6:: Circular Cylinder
* All points are a fixed distance $$r$$ from the $$z$$ axis
- Coordinate System: Cylindrical coordinates - Explanation: This represents a circular surface:
Step 7:: Half Cone
* The height $$z$$ is related to the radial distance $$\sqrt{x^{2} + y^{2}}
- Coordinate System: Cylindrical coordinates - Explanation: This describes a cone-like surface: * Forms a symmetric cone shape extending from the origin
Final Answer
The provided equations represent six fundamental geometric surfaces, each described using specific coordinate systems that best capture their geometric properties.
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