Conic Sections Definition l A conic section is

Conic Sections - Definition l. A conic section is a curve formed by intersecting cone with a plane l There are four types of Conic sections

Conic Sections - Four Types

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

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

2 -Parabola l. 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.

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

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

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

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

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

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

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

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

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

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

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

Parabola l The Parabola that opens to the left and has a vertex at (h, k) has the following characteristics…… l The Standard Form will be: lp is the distance from the vertex of the parabola to the focus or directrix. l This makes the coordinates of the focus (h–p, k). l This makes the equation of the directrix x = h+p. l The makes the axis of symmetry is y=k.

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

Parabola l The Parabola that opens up and has a vertex at (h, k) has the following characteristics…… l The Standard Form will be: lp is the distance from the vertex of the parabola to the focus or directrix. l This makes the coordinates of the focus (h, k + p). l This makes the equation of the directrix y = k–p. l The makes the axis of symmetry is x=h.

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

Parabola l The Parabola that opens down and has a vertex at (h, k) has the following characteristics…… l The Standard Form will be: lp is the distance from the vertex of the parabola to the focus or directrix. l This makes the coordinates of the focus (h, k-p). l This makes the equation of the directrix y=k+p. l This makes the axis of symmetry is x=h.

3 - Ellipse l 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”).

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

Ellipse General Rules l a is the distance from the center to each vertex on the major axis. l b is the distance from the center to each vertex on the minor axis (covertices). l Major axis has a length of 2 a. l Minor axis has a length of 2 b. l Eccentricity(e): e = c/a (The closer e gets to 0, the closer it is to being circular).

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

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

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

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

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

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

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

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

4 - Hyperbola l The set of all points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.

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

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

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

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

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

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

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

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

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

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