Elliptical Galaxies Elliptical Orbit Effect 10 4 Ellipses

Elliptical Galaxies Elliptical Orbit Effect 10. 4 Ellipses Elliptical Orbits Ellipses from Conic Sections By Karen Kidwell

Review. . . What are the two other conic sections we have already discussed? Ø Answer: Circles and Parabolas Is that a parabola? ? Ø What is the equation for Circles? Ø x 2 + y 2 = r 2 Ø What are the new equation for parabolas? Ø x 2 = 4 py or y 2 = 4 px Ø

Teach: Definition: Ø An ellipse is the set of all point P such that the sum of the distances between P and two distinct fixed points, called foci, is a constant. Interactive Demonstrations: http: //www. mathopenref. com/c onstellipse 1. html http: //www. mathopenref. com/ ellipse. html

Vocabulary: Ø Vertices: The line through the foci and intersects the ellipse in two points Ø Major Axis: The line joining the two vertices Ø Center: Midpoint of the Major Axis Ø Co-vertices: The line perpendicular to the major axis at the center and intersects the ellipse in two points Ø Minor Axis: The line segment between the two co-vertices

Diagram: b to b is the minor axis a to a is the major axis the c’s are the foci

Equations Ø The standard form of the equation of an ellipse: Equation Major Axis Vertices Horizont (±a, 0) al ( 0, ±b) Vertical ( 0, ±a) (±b, 0) Co-Vertices The foci of the ellipse are on the major axis, c units from the center where c 2 = a 2 – b 2

How does it work? Draw the ellipse given by 9 x 2 + 16 y 2 = 144. Ø Steps: Ø First you must make the equation equal to 1. See previous slide. Ø Divide by 144 throughout the whole problem Ø Now you have x 2/16 + y 2/9 = 1 Ø What is a and b? (see equation!) Ø Answer a = 4 and b = 3 Ø So, along the horizontal (like x axis, notice it is below the x 2), the length is 2 a, 8 and along the vertical is 2 b, 6, all centered around the origin (0, 0) Ø

Another problem: Write an equation of the ellipse with the given characteristics and center at (0, 0). Ø Vertex: (-4, 0) and Focus: (2, 0) Ø Ø Steps: Ø Ø Ø Ø We know a = what? and c = what? Answer: a = 4 and c = 2 How can we find b? Answer: Use the formula c 2 = a 2 – b 2 What’s b? 22 = 42 – b 2; b 2 = 12; b = 2 3 So the equation is x 2/16 + y 2/12 = 1

Practice: Ø You try: Ø Write in standard form (if not already). Then, identify the vertices, co-vertices and foci of the ellipse: Ø 1. x 2/25 + y 2/16 = 1 Ø 2. 10 x 2 + 25 y 2 = 250 Ø 3. Graph the equation and identify the same above parts. x 2/4 + y 2/49 = 1 Ø Write the equation given Ø 4. Vertex: (0, -7) and Co-vertex: (-1, 0) Ø 5. Vertex: (15, 0) and Focus: (12, 0)

Answers: Ø 1. vertices: (± 5, 0), co-vertices: (0, ± 4), and foci: (± 3, 0) Ø 2. x 2/25 + y 2/10 = 1, vertices: (± 5, 0), covertices: (0, ± 10), and foci: (± 15, 0) Ø 3. Graph of a vertical ellipse; vertices at (0, ± 7) and co-vertices at (± 2, 0) Ø 4. x 2/1 + y 2/49 = 1 Ø 5. x 2/225 + y 2/81 = 1

Apply: Ø Both man-made objects, such as The Ellipse at the White House, and natural phenomena, such as the orbits of planets, involve ellipses.

Solve: Ø 1. A portion of the White House lawn is called The Ellipse. It is 1060 feet long and 890 feet wide. l l A. Write an equation of the Ellipse B. The area of an ellipse is A= πab What is the area of The Ellipse at the White House?

Solve: Ø 2. In its elliptical orbit, Mercury ranges from 46. 04 million kilometers to 69. 86 million kilometers from the center of the sun. The center of the sun is a focus of the orbit. Write an equation of the orbit.

You are on your way to becoming an expert in: Ellipses