Ellipse. Conic Sections

Size: px

Start display at page:

Download "Ellipse. Conic Sections"

Bruce Hart
5 years ago
Views:

1 Ellipse Conic Sections

2 Ellipse The plane can intersect one nappe of the cone at an angle to the axis resulting in an ellipse.

3 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.

4 Finding An Equation Ellipse

5 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).

6 Ellipse - Equation According to the definition. The sum of the distances from the foci to any point on the ellipse is a constant.

7 Ellipse - Equation The distance from the foci to the point (a, 0) is 2a. Why?

8 Ellipse - Equation The distance from (c, 0) to (a, 0) is the same as from (-a, 0) to (-c, 0).

9 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.

10 Ellipse - Equation Therefore, d 1 + d 2 = 2a. Using the distance formula, ( + ) + + ( ) + = 2 x c y x c y a

11 Ellipse - Equation Simplify: ( + ) + + ( ) + = 2 x c y x c y a ( x c) + y = 2 a ( x+ c) + y Square both sides. ( x c) + y = 4a 4 a ( x+ c) + y + ( x+ c) + y Subtract y 2 and square binomials. x 2xc+ c = 4a 4 a ( x+ c) + y + x + 2xc+ c

12 Ellipse - Equation Simplify: x 2xc+ c = 4a 4 a ( x+ c) + y + x + 2xc+ c Solve for the term with the square root. 4xc 4a = 4 a ( x + c) + y xc + a = a ( x + c) + y Square both sides. ( ) 2 xc + a 2 = ( a ( x + c) 2 + y 2 ) 2

13 Ellipse - Equation Simplify: ( ) 2 xc + a 2 = ( a ( x + c) 2 + y 2 ) 2 ( ) x c + 2xca + a = a x + 2xc + c + y x c + 2xca + a = a x + 2xca + a c + a y x c + a = a x + a c + a y Get x terms, y terms, and other terms together. ( ) x c a x a y = a c a

14 Ellipse - Equation Simplify: ( ) x c a x a y = a c a ( c 2 a 2 ) x 2 a 2 y 2 = a 2 ( c 2 a 2 ) Divide both sides by a 2 (c 2 -a 2 ) ( ) 2 2 ay ( ) ( ) ( ) ( ) c a x a c a = a c a a c a a c a x y a c a 2 2 = ( )

15 Ellipse - Equation x y a c a 2 2 = ( ) Change the sign and run the negative through the denominator. 2 2 x y + = a a c ( ) At this point, let s pause and investigate a 2 c 2.

16 Ellipse - Equation d 1 + d 2 must equal 2a. However, the triangle created is an isosceles triangle and d 1 = d 2. Therefore, d 1 and d 2 for the point (0, b) must both equal a.

17 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.

18 Ellipse - Equation We now know.. x y a a c = ( ) and b 2 + c 2 = a 2 b 2 = a 2 c 2 Substituting for a 2 - c 2 x a y + = where c 2 = a 2 b 2 b

19 Ellipse - Equation The equation of an ellipse centered at (0, 0) is. x a y b = where a 2 = b 2 + c 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 = x h y k ( ) ( ) a b where a 2 = b 2 + c 2 and c is the distance from the center to the foci.

20 Ellipse - Graphing = x h y k ( ) ( ) a b where a 2 = b 2 + c 2 and c is the distance from the center to the foci. a c b b c a Vertices are a units in the direction of the major axis and b units in the direction of the minor axis. The foci are c units in the direction of the longer (major) axis.

21 Graph - Example #1 Ellipse Ellipse - Graphing

22 Ellipse - Graphing Graph: 2 2 x 2 y+ 3 + = ( ) ( )

24 Graph - Example #2 Ellipse

25 Ellipse - Graphing Graph: 2 2 5x + 2y + 10x 12y 27 = 0

27 Find An Equation Ellipse

28 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.

30 Ellipse Story Problem A semielliptical arch is to have a span of 100 feet. The height of the arch, at a distance 40 feet from the center is to be 100 feet. Find the height of the arch at its center.

32 Ellipse Story Problem A hall 100 feet in length is to be designed into a whispering gallery. If the foci are located 25 feet from the center, how high will the ceiling be at the center?

34 Assignment: Wksheet #4-7**, 20-23, 33, 38, 46, 47 **Graph and find center, major vertices, minor vertices, and foci Please use graph paper!!

Circles. 1 Page Hannah Province Mathematics Department Southwest Tn Community College

Circles. 1 Page Hannah Province Mathematics Department Southwest Tn Community College Circles 1 Page To Graph a Circle; Graphing Calculator + y = 2 2 First Solve the equation for y: x 4 y = 4-x 2 2 y = ± 4 x 2 2 Graph as two separate equations y = 4 x y = 4 x 1 2 So that the circle doesn't