Conic Sections Conic sections are lines that define

Conic Sections Conic sections are lines that define where a flat plane intersects with a double cone, which consists of two cones that meet at one another’s tip.

Circle

Circle • The Standard Form of a circle with a center at (0, 0) and a radius, r, is……. . center (0, 0) radius = 2

Circles • The Standard Form of a circle with a center at (h, k) and a radius, r, is……. . center (3, 3) radius = 2

Parabolas

What’s in a Parabola • A parabola is the set of all points in a plane such that each point in the set is equidistant from a line called the directrix and a fixed point called the focus.

Why is the focus so important?

Parabola • The Standard Form of a Parabola that opens to the right and has a vertex at (0, 0) is……

Parabola • The Parabola that opens to the right and has a vertex at (0, 0) has the following characteristics…… • p is the distance from the vertex of the parabola to the focus or directrix • This makes the coordinates of the focus (p, 0) • This makes the equation of the directrix x = -p • The makes the axis of symmetry the x-axis (y = 0)

Parabola • The Standard Form of a Parabola that opens to the left and has a vertex at (0, 0) is……

Parabola • The Parabola that opens to the left and has a vertex at (0, 0) has the following characteristics…… • p is the distance from the vertex of the parabola to the focus or directrix • This makes the coordinates of the focus(-p, 0) • This makes the equation of the directrix x = p • The makes the axis of symmetry the x-axis (y = 0)

Parabola • The Standard Form of a Parabola that opens up and has a vertex at (0, 0) is……

Parabola • The Parabola that opens up and has a vertex at (0, 0) has the following characteristics…… • p is the distance from the vertex of the parabola to the focus or directrix • This makes the coordinates of the focus (0, p) • This makes the equation of the directrix y = -p • This makes the axis of symmetry the y-axis (x = 0)

Parabola • The Standard Form of a Parabola that opens down and has a vertex at (0, 0) is……

Parabola • The Parabola that opens down and has a vertex at (0, 0) has the following characteristics…… • p is the distance from the vertex of the parabola to the focus or directrix • This makes the coordinates of the focus (0, -p) • This makes the equation of the directrix y = p • This makes the axis of symmetry the y-axis (x = 0)

Parabola • The Standard Form of a Parabola that opens to the right and has a vertex at (h, k) is……

Parabola • The Parabola that opens to the right and has a vertex at (h, k) has the following characteristics……. . • p is the distance from the vertex of the parabola to the focus or directrix • This makes the coordinates of the focus (h+p, k) • This makes the equation of the directrix x = h – p • This makes the axis of symmetry

Parabola • The Standard Form of a Parabola that opens to the left and has a vertex at (h, k) is……

Parabola • The Parabola that opens to the left and has a vertex at (h, k) has the following characteristics…… • p is the distance from the vertex of the parabola to the focus or directrix • This makes the coordinates of the focus (h – p, k) • This makes the equation of the directrix x = h + p • The makes the axis of symmetry

Parabola • The Standard Form of a Parabola that opens up and has a vertex at (h, k) is……

Parabola • The Parabola that opens up and has a vertex at (h, k) has the following characteristics…… • p is the distance from the vertex of the parabola to the focus or directrix • This makes the coordinates of the focus (h , k + p) • This makes the equation of the directrix y = k – p • The makes the axis of symmetry

Parabola • The Standard Form of a Parabola that opens down and has a vertex at (h, k) is……

Parabola • The Parabola that opens down and has a vertex at (h, k) has the following characteristics…… • p is the distance from the vertex of the parabola to the focus or directrix • This makes the coordinates of the focus (h , k - p) • This makes the equation of the directrix y = k + p • This makes the axis of symmetry

Ellipse • Statuary Hall in the U. S. Capital building is elliptic. It was in this room that John Quincy Adams, while a member of the House of Representatives, discovered this acoustical phenomenon. He situated his desk at a focal point of the elliptical ceiling, easily eavesdropping on the private conversations of other House members located near the other focal point.

What is in an Ellipse? • The set of all points in the plane, the sum of whose distances from two fixed points, called the foci, is a constant. (“Foci” is the plural of “focus”)

Why are the foci of the ellipse important? • The ellipse has an important property that is used in the reflection of light and sound waves. Any light or signal that starts at one focus will be reflected to the other focus. This principle is used in lithotripsy, a medical procedure for treating kidney stones. The patient is placed in a elliptical tank of water, with the kidney stone at one focus. Highenergy shock waves generated at the other focus are concentrated on the stone, pulverizing it.

Why are the foci of the ellipse important? • St. Paul's Cathedral in London. If a person whispers near one focus, he can be heard at the other focus, although he cannot be heard at many places in between.

Ellipse General Rules • • x and y are both squared Equation always equals(=) 1 Equation is always plus(+) a 2 is always the biggest denominator c 2 = a 2 – b 2 c is the distance from the center to each foci on the major axis The center is in the middle of the 2 vertices, the 2 covertices, and the 2 foci.

Ellipse General Rules • • • a is the distance from the center to each vertex on the major axis b is the distance from the center to each vertex on the minor axis (co-vertices) Major axis has a length of 2 a Minor axis has a length of 2 b Eccentricity(e): e = c/a (The closer e gets to 1, the closer it is to being circular)

Ellipse • The standard form of the ellipse with a center at (0, 0) and a horizontal axis is……

Ellipse • The ellipse with a center at (0, 0) and a horizontal axis has the following characteristics…… • Vertices ( a, 0) • Co-Vertices (0, b) • Foci ( c, 0)

Ellipse • The standard form of the ellipse with a center at (0, 0) and a vertical axis is……

Ellipse • The ellipse with a center at (0, 0) and a vertical axis has the following characteristics…… • Vertices (0, a) • Co-Vertices ( b, 0) • Foci (0, c)

Ellipse • The standard form of the ellipse with a center at (h, k) and a horizontal axis is……

Ellipse • The ellipse with a center at (h, k) and a horizontal axis has the following characteristics…… • Vertices (h a , k) • Co-Vertices (h, k b) • Foci (h c , k)

Ellipse • The standard form of the ellipse with a center at (h, k) and a vertical axis is……

Ellipse • The ellipse with a center at (h, k) and a vertical axis has the following characteristics…… • Vertices (h, k a) • Co-Vertices (h b , k) • Foci (h, k c)

Hyperbola The huge chimney of a nuclear power plant has the shape of a hyperboloid, as does the architecture of the James S. Mc. Donnell Planetarium of the St. Louis Science Center.

What is a Hyperbola? • The set of all points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.

Where are the Hyperbolas? • A sonic boom shock wave has the shape of a cone, and it intersects the ground in part of a hyperbola. It hits every point on this curve at the same time, so that people in different places along the curve on the ground hear it at the same time. Because the airplane is moving forward, the hyperbolic curve moves forward and eventually the boom can be heard by everyone in its path.

Hyperbola General Rules • • x and y are both squared Equation always equals(=) 1 Equation is always minus(-) a 2 is always the first denominator c 2 = a 2 + b 2 c is the distance from the center to each foci on the major axis a is the distance from the center to each vertex on the major axis

Hyperbola General Rules • • • b is the distance from the center to each midpoint of the rectangle used to draw the asymptotes. This distance runs perpendicular to the distance (a). Major axis has a length of 2 a Eccentricity(e): e = c/a (The closer e gets to 1, the closer it is to being circular If x 2 is first then the hyperbola is horizontal If y 2 is first then the hyperbola is vertical.

Hyperbola • General Rules • The center is in the middle of the 2 vertices and the 2 foci. • The vertices and the covertices are used to draw the rectangles that form the asymptotes. • The vertices and the covertices are the midpoints of the rectangle • The covertices are not labeled on the hyperbola because they are not actually part of the graph

Hyperbola • The standard form of the Hyperbola with a center at (0, 0) and a horizontal axis is……

Hyperbola • The Hyperbola with a center at (0, 0) and a horizontal axis has the following characteristics…… • • Vertices ( a, 0) Foci ( c, 0) • Asymptotes:

Hyperbola • The standard form of the Hyperbola with a center at (0, 0) and a vertical axis is……

Hyperbola • The Hyperbola with a center at (0, 0) and a vertical axis has the following characteristics…… • • Vertices (0, a) Foci ( 0, c) • Asymptotes:

Hyperbola • The standard form of the Hyperbola with a center at (h, k) and a horizontal axis is……

Hyperbola • The Hyperbola with a center at (h, k) and a horizontal axis has the following characteristics…… • • Vertices (h a, k) Foci (h c, k ) • Asymptotes:

Hyperbola • The standard form of the Hyperbola with a center at (h, k) and a vertical axis is……

Hyperbola • The Hyperbola with a center at (h, k) and a vertical axis has the following characteristics…… • • Vertices (h, k Foci (h, k c) • Asymptotes: a)