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Surfaces)

Conic Sections (review)

Name Form Example Graph

22 Circle (xh− ) +( yk− ) = r2

2 (xh− ) = ayk( − ) or 2 ( yk− ) = axh( − )

22 (xh−−) ( yk) +=1 ab22

22 (xh−−) ( yk) − =1 ab22 or xy= k

The surfaces are surfaces that can be written in the form: Ax222+++++++++= By Cz Dxy Exz Fyz Hx Iy Jz K 0 where ABC, , ,..., K are constants.

There are nine distinct types. These are important, so get familiar with them. You will need to know: (a) The intercepts (the points at which the intersects the coordinate axes). (b) The traces (the intersections with the coordinate planes). (c) The sections (the intersections with planes in general). (d) The center (some have a center; some do not). (e) . (f) Boundedness, unboundedness.

The : centered at the origin and is symmetric about all three coordinate planes.

x2 y2 z2 + + = 1 a2 b2 c2

The of one sheet: unbounded surface, centered at the origin and is symmetric about all three coordinate planes.

x2 y2 z2 + − = 1 a2 b2 c2

The hyperboloid of two sheets: unbounded surface, The surface intersects the coordinate axes only at the two vertices (0, 0, ± c)

x2 y2 z2 z2 x2 y2 + − = −1 or − − = 1 a2 b2 c2 c2 a2 b2

The elliptic : intersects the coordinate axes only at the origin. The surface is unbounded.

x2 y2 + = z2 a2 b2

The elliptic paraboloid: the surface does not extend below the xy-plane; it is unbounded above.

x2 y2 + = z a2 b2

The hyperbolic paraboloid: symmetry about the xz-plane and yz plane. Sections parallel to the xy-plane are ; sections parallel to the other coordinate planes are .

x2 y2 − = z a2 b2

The parabolic : symmetry about the xy-plane

x2 = 4cy

The elliptic cylinder: symmetry about the xy-plane

x2 y2 + = 1 a2 b2

The hyperbolic cylinder: symmetry about the xy-plane

x2 y2 − = 1 a2 b2

Examples:

Examples:

1. Sketch they cylinder 25yz22+ 4− 100= 0

2. Sketch they cylinder zx= 2

3. Identify the surface and find the traces. Then sketch the surface. 9x2 + 4y2 − 36z = 0

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