9.1.1 – Conic Sections; The Ellipse. In math, we define a “conic section” given the equation...
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Transcript of 9.1.1 – Conic Sections; The Ellipse. In math, we define a “conic section” given the equation...
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
022 FEyDxCyAx
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;1
2
2
2
2
b
y
a
x
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;
12
2
2
2
b
y
a
x
12
2
2
2
a
y
b
x
• 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
•
22
222
bac
bac
• 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 1916
22
yx
• Example. Graph the ellipse 1254
22
yx
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.
1)()(
1)()(
2
2
2
2
2
2
2
2
a
ky
b
hx
b
ky
a
hx
• 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 116
)2(
4
)5( 22
yx
• Example. Graph the ellipse 116
)3(
9
22
yx
• Assignment• Pg. 706• 13-20 all• 21-29 odd