<Text-field style="Heading 1" layout="Heading 1">Visualizing the Conic Sections</Text-field>

Conic Sections

Calculus III Lab

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The conic sections can be interpreted as the graphs of second degree equations -- that is equations of the form 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.

The reason that these graphs are called conic sections is that they arise as the intersection of a plane with a right circular cone.

At MathWorld you can find a lot of information about conics sections.

The following command displays a right circular cylinder using cylindrical coordinates (not covered until later)

We shall ignore the syntax and just appreciate the graph:

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JSFH

<Text-field style="Heading 2" layout="Heading 2"> Parabola as a Conic Section</Text-field>

Here we plot the cylinder along with an intersecting plane (gray) with the curve of intersection being a parabola (black curve).

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JSFH

<Text-field style="Heading 2" layout="Heading 2">Ellipse as a Conic Section</Text-field>

The ellipse is generated as the intersection of a cone with a plane that makes a shallower angle with the side of the cone.

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JSFH

<Text-field style="Heading 2" layout="Heading 2">Hyperbola as a Conic Section</Text-field>

The hyperbola is generated as the intersection of a cone with a plane that makes a steeper angle with the side of the cone.

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<Text-field style="Heading 1" layout="Heading 1"> Exercises</Text-field>

Maple's ConicsTutor which is part of the Student[Precalculus] package can be useful for analyzing equations representing the various conic sections. We first load the Precalculus package:

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

Next invoke the Conics Tutor --

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEsQ29uaWNzVHV0b3JGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYjUSFGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZARj1GQA==

Complete the following exercises from your text: page 704-705: 19, 34, 48, 52. Sketch the graphs on the paper given to you along with computations (neatly).

Use ConicsTutor to check your work.

JSFH