Transcript Slide 1
Ellipse
Conic Sections
Ellipse The plane can intersect one nappe of the cone at an angle to the axis resulting in an
ellipse
.
Ellipse - Definition An ellipse is the set of all points in a plane such that the sum of the distances from two points (foci) is a constant. d 1 + d 2 = a constant value.
Finding An Equation
Ellipse
Ellipse - Equation To find the equation of an ellipse, let the center be at (0, 0). The vertices on the axes are at (a, 0), (-a, 0), (0, b) and (0, -b). The foci are at (c, 0) and (-c, 0).
Ellipse - Equation According to the definition. The sum of the distances from the foci to any point on the ellipse is a constant.
Ellipse - Equation The distance from the foci to the point (a, 0) is 2a. Why?
Ellipse - Equation The distance from (c, 0) to (a, 0) is the same as from (-a, 0) to (-c, 0).
Ellipse - Equation The distance from (-c, 0) to (a, 0) added to the distance from (-a, 0) to (-c, 0) is the same as going from (-a, 0) to (a, 0) which is a distance of 2a.
Ellipse - Equation Therefore, d 1 + d 2 = 2a. Using the distance formula, (
x
c
) 2
y
2 ( ) 2
y
2 2
a
Ellipse - Equation Simplify: (
x
c
) 2
y
2 ( ) 2
y
2 2
a
( ) 2
y
2 2
a
(
x
c
) 2
y
2 Square both sides.
( ) 2
y
2 4
a
2 4
a
(
x
c
) 2
y
2 Subtract y 2 and square binomials.
x
2 2
xc
c
2 4
a
2 4
a x c
) 2
y
2 (
x
c
) 2
y
2
x
2 2
xc
c
2
Ellipse - Equation Simplify:
x
2 2
xc
c
2 4
a
2 4
a
(
x
c
) 2
y
2
x
2 2
xc
c
2 Solve for the term with the square root.
4
xc
4
a
2 4
a
(
x
c
) 2
y
2
xc
a
2
a
(
x
c
) 2
y
2 Square both sides.
xc
a
2 2
a
( ) 2
y
2 2
Ellipse - Equation Simplify:
xc
2
x c
2 2 2
x c a
2 2
a
2
xca
2 2
xca
2 (
a a
4 4 ) 2
y
2
a
2 2
a x
2 2
x
2 2
xc
2
xca
2
c
2 2 2
a c y
2 2
a y
2 2 2
x c
a
4 2
a x
2 2 2
a c
2
a y
2 Get x terms, y terms, and other terms together.
2
x c
2 2
a x
2 2
a y
2 2
a c
2
a
4
Ellipse - Equation Simplify: 2
x c
2
c
2 2
a x
2
a
2
x
2 2
a y
2 2
a y
2
a
2
c
2 2
a c
2
a
2
a
4 Divide both sides by a 2 (c 2 -a 2 )
a
2
c
2
c
2
a
2
a
2
x
2
a
2
c
2 2
a
2
a
2
a
2
c
2
c
2
a
2
a
2
x
2
a
2
c
2
y
2
a
2 1
Ellipse - Equation
x
2
a
2
c
2
y
2
a
2 1 Change the sign and run the negative through the denominator.
x
2
a
2
y
2
a
2
c
2 1 At this point, let’s pause and investigate a 2 – c 2 .
Ellipse - Equation d 1 + d 2 must equal 2a. However, the triangle created is an isosceles triangle and d 1 d 2 = d 2 . Therefore, d 1 for the point (0, b) must both equal “a”.
and
Ellipse - Equation This creates a right triangle with hypotenuse of length “a” and legs of length “b” and “c”. Using the pythagorean theorem, b 2 + c 2 = a 2 .
Ellipse - Equation We now know…..
x
2
a
2
a
2
y
2
c
2 1 and b 2 + c 2 = a 2 b 2 = a 2 – c 2 Substituting for a 2 - c 2
x
2
a
2
y b
2 2 1 where c 2 = |a 2 – b 2 |
Ellipse - Equation The equation of an ellipse centered at (0, 0) is ….
x
2
a
2
y b
2 2 1 where c 2 = |a 2 – b 2 | and c is the distance from the center to the foci.
Shifting the graph over h units and up k units, the center is at (h, k) and the equation is
y
k
2 1
a
2
b
2 where c 2 = |a 2 – b 2 | and c is the distance from the center to the foci.
Ellipse - Graphing
y
k
1 where c 2 = |a 2 – b 2 | and c is the distance from the
a
2
b
2 2 center to the foci.
b Vertices are “a” units in the x direction and “b” units in the y direction.
a c c a b The foci are “c” units in the direction of the longer (major) axis.
Graph - Example #1
Ellipse
Ellipse - Graphing Graph:
x
2 16
y
3 2 1 25 Center: (2, -3) Distance to vertices in x direction: 4 Distance to vertices in y direction: 5 Distance to foci: c 2 =|16 - 25| c 2 = 9 c = 3
Ellipse - Graphing Graph:
x
2 16
y
3 2 1 25 Center: (2, -3) Distance to vertices in x direction: 4 Distance to vertices in y direction: 5 Distance to foci: c 2 =|16 - 25| c 2 = 9 c = 3
Graph - Example #2
Ellipse
Ellipse - Graphing Graph: 5
x
2 2
y
2 10
x
12
y
27 0 Complete the squares.
5 5 5 5
x
x x x
2 2 2 10 1 2 2 2
x x x
2 ??
y
2 2 12
y
2 2
y
3
y
2 2 6
y
y
6 50
y
27 9 ??
x
10 1
y
25 3 2 1 27
Ellipse - Graphing Graph:
x
10 1
y
25 3 2 1 Center: (-1, 3) -5 -2 -4 8 6 4 2 5 Distance to vertices in x direction: 10 Distance to vertices in y direction: 5 Distance to foci: c 2 =|25 - 10| c 2 = 15 c = 15
Find An Equation
Ellipse
Ellipse – Find An Equation Find an equation of an ellipse with foci at (-1, -3) and (5, -3). The minor axis has a length of 4. The center is the midpoint of the foci or (2, -3). The minor axis has a length of 4 and the vertices must be 2 units from the center. Start writing the equation.
Ellipse – Find An Equation
y
k
2 1
a
2
b
2
x
2
a
2 c 2 = |a 2
y
4 3 2 – b 2 |. Since 1 the major axis is in the x direction, a 2 > 4 9 = a 2 – 4 a 2 = 13 Replace a 2 in the equation.
Ellipse – Find An Equation The equation is:
x
2 13
y
3 2 1 4
Ellipse – Table
a
2
y
k
2
b
2 1 Center: Vertices: (h, k) Foci: If a 2 > b 2 If b 2 > a 2 c 2 = |a 2 – b 2 |
c
b