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