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 x 2 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 coordinates of the focus? –What is the equation of the directrix? (h, k)

When the Vertex Is (h, k) Standard form of equation for horizontal axis of symmetry Consider –What are the coordinates of the focus? –What is the equation of 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 Ellipse Part A

Ellipse Another conic section formed by a plane intersecting a cone 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 with dist. 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 parallel to x-axis equation can be shown to be

Ellipses with Center at (h,k) When major axis parallel to 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