Conic Sections The Parabola
Introduction • Consider a ___________ being intersected with a __________
Introduction • We will consider various conic sections and how they are described analytically • Parabolas • Hyperbolas • Ellipses • Circles
Parabola • Definition • A set of points on the plane that are equidistant from • A fixed line (the ____________) and • A fixed point (the __________) not on the directrix
Parabola • Note the line through the focus, perpendicular to the directrix • Axis of symmetry • Note the point midway between the directrix and the focus • ______________
Equation of Parabola • Let the vertex be at (0, 0) • Axis of symmetry be y-axis • Directrix be the line y = -p (where p > 0) • Focus is then at (0, p) • For any point (x, y) on the parabola
Equation of Parabola • Setting the two distances equal to each other • What happens if p < 0? • What happens if we have . . . simplifying . . .
Working with the Formula • Given the equation of a parabola • y = ½ x2 • Determine • The directrix • The focus • Given the focus at (-3,0) and the fact that the vertex is at the origin • Determine the equation
When the Vertex Is (h, k) • Standard form of equation for vertical axis of symmetry • Consider • What are the coordinatesof the focus? • What is the equationof the directrix? (h, k)
When the Vertex Is (h, k) • Standard form of equation for horizontal axis of symmetry • Consider • What are the coordinatesof the focus? • What is the equationof the directrix? (h, k)
Try It Out • Given the equations below, • What is the focus? • What is the directrix?
Another Concept • Given the directrix at x = -1 and focus at (3,2) • Determine the standard form of the parabola
Assignment • See Handout • Part A 1 – 33 odd • Part B 35 – 43 all
Conic Sections The EllipsePart A
Ellipse • Another conicsection formedby a plane intersecting acone • Ellipse formed when
Definition of Ellipse • Set of all points in the plane … • ___________ of distances from two fixed points (foci) is a positive _____________
Definition of Ellipse • Definition demonstrated by using two tacks and a length of string to draw an ellipse
Graph of an Ellipse Note various parts of an ellipse
Deriving the Formula • Note • Why? • Write withdist. formula • Simplify
Major Axis on y-Axis • Standard form of equation becomes • In both cases • Length of major axis = _______ • Length of __________ axis = 2b
Using the Equation • Given an ellipse with equation • Determine foci • Determine values for a,b, and c • Sketch the graph
Find the Equation • Given that an ellipse … • Has its center at (0,0) • Has a minor axis of length 6 • Has foci at (0,4) and (0,-4) • What is the equation?
Ellipses with Center at (h,k) • When major axis parallelto x-axis equation can be shown to be
Ellipses with Center at (h,k) • When major axis parallelto y-axis equation can be shown to be
Find Vertices, Foci • Given the following equations, find the vertices and foci of these ellipses centered at (h, k)
Find the Equation • Consider an ellipse with • Center at (0,3) • Minor axis of length 4 • Focci at (0,0) and (0,6) • What is the equation?
Assignment • Ellipses A • 1 – 43 Odd
