Horizontal Plane? Diagonal Plane (less steep than the cone) Diagonal Plane (parallel to the slope of the cone) Vertical Plane? (steeper than the slope of the cone) CONIC SECTIONS If we pass a plane through an infinite cone, the intersection will be… Circle Ellipse Parabola Hyperbola Lesson: ____ Section: 9.1 Conic Sections

In rhetoric, "hyperbolic" speech is the kind that goes beyond the facts, "elliptic speech" falls short of them, while a "parable" is a story that is meant to fit the facts. The term parabola comes from Greek para “with, alongside, nearby, right up to," and -bola, from the verb ballein "to cast, to throw." Understandably, parallel and many of its derivatives start with the same root. The word parabola may mean "thrown parallel" Ellipse Parabola Hyperbola Parabola - parallel to the slope of the cone Ellipse - “falls short” of the slope of the cone. Hyperbola – “goes beyond” the slope of the cone

The Equation of a Circle: This is the equation of all points (x,y) whose distance from the center (h,k) is r. (h, k) is the center of the circle r is the radius (x, y) (h, k) r

Where did this formula come from? Ex 3 : C (2,-7), r = 3 Equation? In the equation of a circle, which letters represent constants? which represent variables? h, k, r x, y

Ex 4 : C (-2,4) with a point on the circle at (4,12) Equation?

Write the equation in standard (completed square) form, then graph the circle.

What is the locus of points equidistant from a fixed line and a fixed point not on the line? What is the locus of points equidistant from a fixed point? What is the locus of points equidistant from two unique fixed points? Locus means “position” or “the set of all points that satisfy the criteria.”

Definition of a PARABOLA: The set of all points (x, y) that are equidistant from a fixed line (directrix) and a fixed point (focus) not on the line. directrix F V

The Equation of a Parabola in Standard Form:  vertical axis p>0 p<0  horiz. axis What is p? p is the distance between the focus and the vertex. F V p

Find the vertex, focus, directrix, & sketch the graph of the parabola V: F: directrix:

Write the equation of the parabola with vertex (3,1) and focus (6,1).

Write the equation of the parabola with a focus at (4,0) and a directrix at x = –4

Convert the equation to standard form.

Definition of an ELLIPSE: The set of all points (x, y) the sum of whose distances from two fixed points (foci) is constant. Minor Axis Major Axis F1F1 F2F2 d1d1 d2d2 d 1 + d 2 is constant

The standard form equation of an ellipse with center (h,k) Major Minor a b b a b The foci lie on the major axis c units from the center, where and axes of length 2a and 2b

Sketch the graph of the relation

Write the equation of the ellipse with vertices (0,2) and (4,2) and a minor axis of length 2.

Eccentricity = a c Not eccentric Highly eccentric

F F Definition of an HYPERBOLA: The set of all points (x, y) the difference of whose distances from two fixed points (foci) is constant. F1F1 F2F2 |d 1 - d 2 | is constant  d1d1 d2d2 Transverse Axis V V C(h,k) c a

The standard form equation of a hyperbola with center (h,k) Horizontal Transverse Axis Vertical Transverse Axis The foci lie on the transverse axis c units from the center, where

Write the equation of a hyperbola with vertices (4,1) and (4,9) and foci at (4,0) and (4,10)

Asymptotes of a Hyperbola: All hyperbolas have 2 asymptotes that intersect at the center and pass through a rectangle of dimensions 2a by 2b. a b Transverse Axis Conjugate Axis 2a 2b

Asymptotes for a Horizontal Transverse Axis Asymptotes for a Vertical Transverse Axis rise / run Asymptotes of a Hyperbola a b

Sketch the graph of the relation What are the equations of the asymptotes?

Classifying Conics: See p.685 for applications to radar and comets