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**9.1.1 – Conic Sections; The Ellipse**

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**In math, we define a “conic section” given the equation**

From the above equation, we have several different types of conics we may define The first, is known as an ellipse If AC > 0, then the conic is an ellipse

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**Ellipse There are several properties and features to an ellipse**

There exist two points in the plane for which their sum of distances, d1 and d2, to two foci, is a fixed constant

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Equation of an Ellipse In terms of Ellipses, we may have one of two types Major Axis = line segment extending from one end (extreme) of an ellipse to the other and passing through the two foci and center Length = 2a Minor Axis = axis perpendicular to major axis Length = 2b Centered at Origin;

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Origin Equations If an ellipse is centered at the origin, and the major axis is horizontal, then the equation is; If an ellipse is centered at the origin, and the major axis is vertical, then the equation is;

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**How do I tell if the major axis is vertical or horizontal?**

If the coefficient below y is GREATER than the coefficient below x, then the graph is stretched vertically; major axis would be vertical If the coefficient below x is GREATER than the coefficient below y, then the graph is stretch horizontally; major axis would be horizontal Major Axis Length = 2a Minor Axis Length = 2b

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Foci To identify the foci, or the points that form a constant, we can use the following formula

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**To graph a standard ellipse, we will do the following**

1) Determine major axis (for reference) 2) Find x and y intercepts 3) Plot the 4 “vertices” 4) Solve for foci and plot them

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**Example. Graph the ellipse**

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**Example. Graph the ellipse**

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Center NOT at origin Just like most other cases, similar to circles, not all ellipses will be centered at the origin The new form is given as; where (h,k) is the center.

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When graphing with a different center, it’s best to determine the lengths of the major and minor axis Just remember, major corresponds to largest coefficient; minor corresponds to smallest coefficient Length of axis starts from center

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**Example. Graph the ellipse**

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**Example. Graph the ellipse**

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Assignment Pg. 706 13-20 all 21-29 odd

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