Questions over Assignment  3R- One more thing we need to do on 8, 9, & 10

Conic Sections- Ellipses (Day 1) A conic section is the intersection of a plane and a cone

 Ellipse: Given two points F and F' (called the foci), the ellipse is the set of points whose sum of distances to the foci is constant.

 Center: A point inside the ellipse which is the midpoint of the line segment linking the two foci. The intersection of the major and minor axes.

 Major/Minor Axis: The longest and shortest diameters of an ellipse.

 Focus Points/Foci: The points from which the ellipse is defined. “The ellipse is the set of points whose sum of distances to the foci is constant.” P is a point on the Ellipse

 Vertex: Where the Major Axis intersects with the ellipse  Co-Vertex: Where the Minor axis intersects with the ellipse

Equations of Ellipses (Horizontal, Centered at the Origin) Foci: (-c,0) and (c,0) Vertices: (-a,0) and (a,0) Co-vertices: (0,-b) and (0,b)

Equations of Ellipses (Vertical, Centered at the Origin) Foci: (0, -c) and (0, c) Vertices: (0, -a) and (0, a) Co-vertices: (-b,0) and (-b,0)

Equation for the Foci Yes, this comes from Pythagorean Theorem. No, the variables used are not the same. No, C is not the hypotenuse. Yes, that can be confusing. No, you do not need to remember where this formula comes from. Yes, you should be able to use it, though.

Example Problem: >Sketch the graph and find the vertices, co-vertices, and foci points for: x² + 4y² = 16 Solution: First put the equation in the correct form by dividing everything by 16: x²/16 + y²/4 = 1 Since the larger value is under x, the ellipse has a horizontal major axis, so a² = 16 and b² = 4. >The values are a = 4, b = 2. >To find c, use c 2 =a 2 -b 2, c 2 =16 – 4 c= 3.5 Center at (0, 0) Vertices: (4, 0) and (-4, 0) Co-Vertices: (0, 2) and (0, -2) Foci: (3.5, 0) and (-3.5, 0)

Center at (0, 0) Vertices: (4, 0) and (-4, 0) End Co-Vertices: (0, 2) and (0, -2) Foci: (3.5, 0) and (-3.5, 0)

Homework…