 9.1.1 – Conic Sections; The Ellipse

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9.1.1 – Conic Sections; The Ellipse

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

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

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;

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;

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

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

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

Example. Graph the ellipse

Example. Graph the ellipse

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.

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

Example. Graph the ellipse

Example. Graph the ellipse

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