ellipses - formulas and graphs
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TRANSCRIPT
Algebra IIAlgebra II
Equations of EllipsesEquations of Ellipses
Basic EquationsBasic Equations
Center is at (Center is at (hh, , kk)) Remember to change signsRemember to change signs
Major axis is determined by Major axis is determined by aa Always the larger denominatorAlways the larger denominator Association with Association with xx or or yy determines direction determines direction Length is 2Length is 2aa or | or |aa| units each direction from the | units each direction from the
centercenter The points on the major axis are vertices The points on the major axis are vertices
1)()(
2
2
2
2
b
ky
a
hx1
)()(2
2
2
2
a
ky
b
hx
Basic Equations Basic Equations (cont.)(cont.)
Minor axis is determined by Minor axis is determined by bb Perpendicular to major axisPerpendicular to major axis Length is 2Length is 2bb or | or |bb| units each direction from | units each direction from
the centerthe center The points on the minor axis are CO-The points on the minor axis are CO-
verticesvertices
1)()(
2
2
2
2
b
ky
a
hx1
)()(2
2
2
2
a
ky
b
hx
Basic Equations Basic Equations (cont.)(cont.)
Foci are |c| units each direction from the Foci are |c| units each direction from the center on the major axiscenter on the major axis
Foci are determined by the equation Foci are determined by the equation aboveabove
222 cba
Features of an EllipseFeatures of an Ellipse
Include sketch of graph for all!Include sketch of graph for all!
Put in standard form by dividing to get “=1”Put in standard form by dividing to get “=1”
1494
196
196
196
4
196
49
196449
22
22
22
yx
yx
yx
Features of an Ellipse Features of an Ellipse (cont.)(cont.)
1494
22
yx
1)()(
2
2
2
2
a
ky
b
hx
Since nothing is with the x or y Since nothing is with the x or y h = 0h = 0 k = 0k = 0 the center is at the originthe center is at the origin
Features of an Ellipse Features of an Ellipse (cont.)(cont.)
1494
22
yx
1)()(
2
2
2
2
a
ky
b
hx
larger denominator determines the equationlarger denominator determines the equation larger denominator is always larger denominator is always aa22
aa22 = 49 = 49 aa = = ±7±7 aa is with is with yy major axis is in major axis is in yy-direction-direction bb = = ±2±2
-8
-6
-4
-2
0
2
4
6
8
-3 -2 -1 0 1 2 3
Features of an Ellipse Features of an Ellipse (cont.)(cont.)
major axis is in major axis is in yy-direction-direction MeasureMeasure a a = = ±7 from ±7 from
center in center in yy-direction-direction MeasureMeasure b b = = ±2 from ±2 from
center in center in xx-direction-direction
b
a
-8
-6
-4
-2
0
2
4
6
8
-3 -2 -1 0 1 2 3
Features of an Ellipse Features of an Ellipse (cont.)(cont.)
Sketch the graphSketch the graph Calculate the fociCalculate the foci
aa22 – b – b22 = c = c22
49 – 4 = c49 – 4 = c22
45 = c45 = c22
±6.7 ±6.7 c c
Plot foci on major axisPlot foci on major axis
c
Developing Equation for EllipseDeveloping Equation for Ellipse
Center (0, 0)Center (0, 0) Co-vertex (0, 4)Co-vertex (0, 4) Vertex (10, 0)Vertex (10, 0)
Must be two vertices Must be two vertices also (–10, 0) also (–10, 0) Point is on Point is on xx-axis means this is the major axis-axis means this is the major axis
Determines which formula to useDetermines which formula to use aa must be with the must be with the xx
1)()(
2
2
2
2
b
ky
a
hx
Developing Equation for EllipseDeveloping Equation for Ellipse(cont)(cont)
Vertex (10, 0) & Vertex (10, 0) & (–10, 0)(–10, 0) aa is distance from center to vertex is distance from center to vertex aa = 10 = 10 aa22 = 100 = 100
Co-vertex (0, 4) & (0, -4)Co-vertex (0, 4) & (0, -4) bb is distance from center to co-vertex is distance from center to co-vertex bb = 4 = 4 bb22 = 16 = 16
Developing Equation for EllipseDeveloping Equation for Ellipse(cont)(cont)
Center (0, 0) Center (0, 0) h = 0h = 0 k = 0k = 0
Plug-inPlug-in a a22 & b & b22 aa22 = 100 = 100 bb22 = 16 = 16
116100
22
yx
Developing Equation for EllipseDeveloping Equation for Ellipse(cont)(cont)
For problems giving focusFor problems giving focus Use: Use: aa22 – b – b22 = c = c2 2 to solve for the missing to solve for the missing
valuevalue Remember the focus is cRemember the focus is c