10.4 Ellipses Elliptical Orbits Elliptical Galaxies Elliptical Orbit Effect Ellipses from Conic Sections By Karen Kidwell

Review. . . What are the two other conic sections we have already discussed? Answer: Circles and Parabolas What is the equation for Circles? x2 + y2 = r2 What are the new equation for parabolas? x2 = 4py or y2 = 4px Is that a parabola??

Teach: Definition: 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/constellipse1.html http://www.mathopenref.com/ellipse.html

Vocabulary: Vertices: The line through the foci and intersects the ellipse in two points Major Axis: The line joining the two vertices Center: Midpoint of the Major Axis Co-vertices: The line perpendicular to the major axis at the center and intersects the ellipse in two points 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 The standard form of the equation of an ellipse: Horizontal Equation Major Axis Vertices Co-Vertices Horizontal (±a, 0) ( 0, ±b) Vertical ( 0, ±a) (±b, 0) The foci of the ellipse are on the major axis, c units from the center where c2 = a2 – b2

How does it work? Draw the ellipse given by 9x2 + 16y2 = 144. Steps: First you must make the equation equal to 1. See previous slide. Divide by 144 throughout the whole problem Now you have x2/16 + y2/9 = 1 What is a and b? (see equation!) Answer a = 4 and b = 3 So, along the horizontal (like x axis, notice it is below the x2), the length is 2a, 8 and along the vertical is 2b, 6, all centered around the origin (0, 0)

Another problem: Steps: Write an equation of the ellipse with the given characteristics and center at (0, 0). Vertex: (-4, 0) and Focus: (2,0) Steps: We know a = what? and c = what? Answer: a = 4 and c = 2 How can we find b? Answer: Use the formula c2 = a2 – b2 What’s b? 22 = 42 – b2; b2 = 12; b = 23 So the equation is x2/16 + y2/12 = 1

Practice Practice: You try: Write in standard form (if not already). Then, identify the vertices, co-vertices and foci of the ellipse: 1. x2/25 + y2/16 = 1 2. 10x2 + 25y2 = 250 3. Graph the equation and identify the same above parts. x2/4 + y2/49 = 1 Write the equation given 4. Vertex: (0, -7) and Co-vertex: (-1, 0) 5. Vertex: (15, 0) and Focus: (12, 0)

Answers: 1. vertices: (±5, 0), co-vertices: (0, ±4), and foci: (±3, 0) 2. x2/25 + y2/10 = 1, vertices: (±5, 0), co-vertices: (0, ±10), and foci: (±15, 0) 3. Graph of a vertical ellipse; vertices at (0,±7) and co-vertices at (±2, 0) 4. x2/1 + y2/49 = 1 5. x2/225 + y2/81 = 1

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

Solve: 1. A portion of the White House lawn is called The Ellipse. It is 1060 feet long and 890 feet wide. 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: 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.

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