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<td><img width="125" height="180" src="https://math2.org/math/graphs/conics/cone-cir.jpg" alt="circle conic" /></td>
<td><img width="125" height="180" src="https://math2.org/math/graphs/conics/cone-ell.jpg" alt="ellipse conic" /></td>
<td><img width="125" height="180" src="https://math2.org/math/graphs/conics/cone-par.jpg" alt="parabola conic" /></td>
<td><img width="125" height="180" src="https://math2.org/math/graphs/conics/cone-hyp.jpg" alt="hyperbola conic" /></td>
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<td>Circle<br />
<img width="125" height="110" src="https://math2.org/math/graphs/conics/g-circle.gif" alt="graph circle (horiz.)" /></td>
<td>Ellipse (h)<br />
<img width="125" height="110" src="https://math2.org/math/graphs/conics/g-ellx.gif" alt="graph ellipse (horiz.)" /></td>
<td>Parabola (h)<br />
<img width="125" height="110" src="https://math2.org/math/graphs/conics/g-parx.gif" alt="graph parabola (horiz.)" /></td>
<td>Hyperbola (h)<br />
<img width="125" height="110" src="https://math2.org/math/graphs/conics/g-hypx.gif" alt="graph hyperbola (horiz.)" /></td>
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<td width="125"><strong><em>Definition:</em></strong><br />
A conic section is the intersection of a plane and a cone.</td>
<td>Ellipse (v)<br />
<img width="125" height="110" src="https://math2.org/math/graphs/conics/g-elly.gif" alt="graph ellipse (vert.)" /></td>
<td>Parabola (v)<br />
<img width="125" height="110" src="https://math2.org/math/graphs/conics/g-pary.gif" alt="graph parabola (vert.)" /></td>
<td>Hyperbola (v)<br />
<img width="125" height="110" src="https://math2.org/math/graphs/conics/g-hypy.gif" alt="graph hyperbola (vert.)" /></td>
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<br />
By changing the angle and location of intersection, we can produce a circle, ellipse, parabola or hyperbola; or in the special case when the plane touches the vertex: a point, line or 2 intersecting lines.
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<td><img width="125" height="180" src="https://math2.org/math/graphs/conics/cone-pnt.jpg" alt="point conic" /></td>
<td><img width="125" height="180" src="https://math2.org/math/graphs/conics/cone-lin.jpg" alt="line conic" /></td>
<td><img width="125" height="180" src="https://math2.org/math/graphs/conics/cone-li2.jpg" alt="double line conic" /></td>
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<td>Point<br />
<img width="125" height="110" src="https://math2.org/math/graphs/g-point.gif" alt="graph point conic" /></td>
<td>Line<br />
<img width="125" height="110" src="https://math2.org/math/graphs/g-mxpb.gif" alt="graph line conic" /></td>
<td width="125" valign="top">Double Line<br />
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<td><font color="#800000"><strong>The General Equation for a Conic Section:</strong></font><br />
Ax<sup>2</sup> + Bxy + Cy<sup>2</sup> + Dx + Ey + F = 0</td>
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<div><font color="#800000"><strong>The type of section can be found from the sign of: B<sup>2</sup> - 4AC</strong></font>
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<td>If B<sup>2</sup> - 4AC is...</td>
<td>then the curve is a...<br />
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<td> < 0</td>
<td>ellipse, circle, point or no curve.<br />
</td>
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<td> = 0</td>
<td>parabola, 2 parallel lines, 1 line or no curve.<br />
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<td> > 0</td>
<td>hyperbola or 2 intersecting lines.<br />
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<div><strong>The Conic Sections.</strong> For any of the below with a center (j, k) instead of (0, 0), replace each <u>x</u> term with (x-j) and each <u>y</u> term with (y-k).
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<td> </td>
<td><font color="#800000"><strong>Circle</strong></font></td>
<td><font color="#800000"><strong>Ellipse</strong></font></td>
<td><font color="#800000"><strong>Parabola</strong></font></td>
<td><font color="#800000"><strong>Hyperbola</strong></font></td>
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<td>Equation (horiz. vertex):</td>
<td>x<sup>2</sup> + y<sup>2</sup> = r<sup>2</sup></td>
<td>x<sup>2</sup> / a<sup>2</sup> + y<sup>2</sup> / b<sup>2</sup> = 1</td>
<td>4px = y<sup>2</sup></td>
<td>x<sup>2</sup> / a<sup>2</sup> - y<sup>2</sup> / b<sup>2</sup> = 1</td>
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<td>Equations of Asymptotes:</td>
<td> </td>
<td> </td>
<td> </td>
<td>y = ± (b/a)x</td>
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<td>Equation (vert. vertex):</td>
<td>x<sup>2</sup> + y<sup>2</sup> = r<sup>2</sup></td>
<td>y<sup>2</sup> / a<sup>2</sup> + x<sup>2</sup> / b<sup>2</sup> = 1</td>
<td>4py = x<sup>2</sup></td>
<td>y<sup>2</sup> / a<sup>2</sup> - x<sup>2</sup> / b<sup>2</sup> = 1</td>
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<td>Equations of Asymptotes:</td>
<td> </td>
<td> </td>
<td> </td>
<td>x = ± (b/a)y</td>
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<td>Variables:</td>
<td>r = circle radius</td>
<td>a = major radius (= 1/2 length major axis)<br />
b = minor radius (= 1/2 length minor axis)<br />
c = distance center to focus</td>
<td>p = distance from vertex to focus (or directrix)</td>
<td>a = 1/2 length major axis<br />
b = 1/2 length minor axis<br />
c = distance center to focus</td>
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<td>Eccentricity:</td>
<td>0</td>
<td> </td>
<td>c/a</td>
<td>c/a</td>
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<td>Relation to Focus:</td>
<td>p = 0</td>
<td>a<sup>2</sup> - b<sup>2</sup> = c<sup>2</sup></td>
<td>p = p</td>
<td>a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></td>
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<td>Definition: is the locus of all points which meet the condition...</td>
<td>distance to the origin is constant</td>
<td>sum of distances to each focus is constant</td>
<td>distance to focus = distance to directrix</td>
<td>difference between distances to each foci is constant</td>
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