WHAT IS THE FORMULA OF X2 + Y2 + Z2 CLASS 8 MATHS CBSE

     

Surfaces & Contour Plots

Part 2: Quadric Surfaces

Quadric surfaces are the graphs of quadratic equations in three Cartesian variables in space. Lượt thích the graphs of quadratics in the plane, their shapes depend on the signs of the various coefficients in their quadratic equations.

Spheres and Ellipsoids

A sphere is the graph of an equation of the khung x2+y2+z2=p2 for some real number p. The radius of the sphere is p. (see the figure below). Ellipsoids are the graphs of equations of the size ax2+by2+cz2=p2, where a, b, and care all positive. In particular, a sphere is a very special ellipsoid for which a, b, & care all equal.

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Plot the graph of x2+y2+z2=4 in your worksheet in Cartesian coordinates. Then choose different coefficients in the equation, & plot a non-spherical ellipsoid.

What curves vì chưng you find when you intersect a sphere with a plane perpendicular to one of the coordinate axes? What vày you find for an ellipsoid?Paraboloids

Surfaces whose intersections with planes perpendicular to lớn any two of the coordinate axes are parabolas in those planes are called paraboloids. An example is shown in the figure below -- this is the graph of z=x2+y2.

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Make your own plot of this surface in your worksheet, and rotate the plot to see it from various perspectives. Follow the suggestions in the worksheet. What are the intersections of the surface with planes of the form z=c, for some constant c?

Show that the intersections of this surface with planes perpendicular to the x- and y-axes are parabolas. Change the equation to lớn z=3x2+y2, & plot again. How does the surface change? In particular, what happens khổng lồ the curves of intersection with horizontal planes. The surface in the following figure is the graph of z=x2-y2. In this case, the intersections with planes perpendicular to the x- and y-axes are still parabolas, but the two sets of parabolas differ in the direction in which they point. For reasons we will see, this surface is called a hyperbolic paraboloid -- and, for obvious reasons, it is also called a "saddle surface."

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Make your own plot of this hyperbolic paraboloid in your worksheet, & rotate the plot khổng lồ see it from various perspectives. Follow the suggestions in the worksheet. What are the intersections of the surface with planes of the size z=c, for some constant c? Explain both parts of the name.

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Hyperboloids

Hyperboloids are the surfaces in three-dimensional space analogous lớn hyperbolas in the plane. Their defining characteristic is that their intersections with planes perpendicular to any two of the coordinate axes are hyperbolas. There are two types of hyperboloids -- the first type is illustrated by the graph of x2+y2-z2=1, which is shown in the figure below. As the figure at the right illustrates, this shape is very similar khổng lồ the one commonly used for nuclear nguồn plant cooling towers. (Source: EPA"s Response khổng lồ Three Mile Island Incident.)

This surface is called a hyperboloid of one sheet because it is all "connected" in one piece. (We will get to lớn the other case presently.)

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Make your own plot of this surface in your worksheet, & rotate the plot lớn see it from various perspectives. Follow the suggestions in the worksheet. What are the intersections of the surface with planes of the form z=c, for some constant c?

Show that the intersections of this surface with planes perpendicular to lớn the x- và y-axes are hyperbolas. The other type is the hyperboloid of two sheets, & it is illustrated by the graph of x2-y2-z2=1, shown below.

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Make your own plot of this surface in your worksheet, & rotate the plot khổng lồ see it from various perspectives. Follow the suggestions in the worksheet. What are the intersections of the surface with planes of the size z=c, for some constant c?

Show that the intersections of these two surfaces with the appropriate coordinate planes are hyperbolas.In each of these examples, the intersections of the surface with a family of planes tells us a great khuyễn mãi giảm giá about the structure of the surface. We will return to lớn this theme in Part 6, when we look at contour lines.

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