conic sections curves with second degree equations

Conic SectionsConic Sections

Curves with Curves with second second degree degree EquationsEquations

Conic ShapesConic Shapes

Conic shapes are obtained by “slicing a Conic shapes are obtained by “slicing a cone” or a “double cone” intersecting at cone” or a “double cone” intersecting at the vertexthe vertex

Different “slices” will obtain different Different “slices” will obtain different curvescurves

The 4 basic curves are : parabola, circle, The 4 basic curves are : parabola, circle, ellipse, hyperbolaellipse, hyperbola


ParabolaParabola This shape is obtainedThis shape is obtained by “slicing a cone”by “slicing a cone” at an angle to at an angle to the “axis” of the “axis” of the conesthe cones


CircleCircle This shape is obtainedThis shape is obtained

by “slicing a cone”by “slicing a cone”

perpendicular toperpendicular to

the axis of the the axis of the

conescones


EllipseEllipse This shape is obtainedThis shape is obtained

by “slicing a cone”by “slicing a cone”

at an angle throughat an angle through

the axis of the conethe axis of the cone


Hyperbola Hyperbola This shape is obtained byThis shape is obtained by

“ “slicing both cones”slicing both cones”

in a slice parallel to in a slice parallel to

to the axis of the to the axis of the

conescones

ParabolaParabola Definition: Set of all points that are Definition: Set of all points that are

equidistant from a given point (equidistant from a given point (focusfocus) ) and a given line (and a given line (directrixdirectrix))

The The vertexvertex is exactly is exactly ½ way between the½ way between the

focus and directrix.focus and directrix. The parabola curves The parabola curves around the focusaround the focus

ParabolaParabola Graphing a ParabolaGraphing a Parabola The simple equation is:The simple equation is:

y = y = 11//(4p)(4p)xx22 or xor x22 = 4py = 4py

““pp” is the distance” is the distance from the vertex tofrom the vertex to either the focus either the focus or directrixor directrix

ParabolaParabola The parabola with the equation of y = The parabola with the equation of y =

11//88xx22 has the following points on its graph: has the following points on its graph: x yx y

-3 1.125-3 1.125

-2 .5-2 .5 -1 .125-1 .125 0 00 0 1 .1251 .125 2 .52 .5 3 1.1253 1.125

ParabolaParabola The parabola with the equation of The parabola with the equation of

y = y = 11//88xx22 has a vertex at the point has a vertex at the point

(0,0). In the equation(0,0). In the equation p = 2.p = 2. This means theThis means the

focusfocus is 2 units is 2 units above the vertexabove the vertex or at the point (0,2).or at the point (0,2).

ParabolaParabola The parabola with the equation of y = The parabola with the equation of y =

11//88xx22 has a vertex at the point has a vertex at the point

(0,0). In the equation(0,0). In the equation p = 2.p = 2. This means theThis means the

directrixdirectrix is is 2 units below2 units below the vertex and is thethe vertex and is the line with the equationline with the equation y = -2y = -2

ParabolaParabola

If the equation has If the equation has xx22, then it is a , then it is a veritcalveritcal parabola. parabola.

If the equation has If the equation has yy22, then it is a , then it is a horizontalhorizontal parabola. parabola.

ParabolaParabola

If If 11//(4p)(4p) is positive, then the parabola is positive, then the parabola

is going in a positive direction (up if is going in a positive direction (up if vertical, right if horizontal).vertical, right if horizontal).

If If 11//(4p)(4p) is negative, then the parabola is negative, then the parabola

is going in a negative direction is going in a negative direction (down if vertical, left if horizontal)(down if vertical, left if horizontal)

Parabola ExampleParabola Example

For the parabola with the equation:For the parabola with the equation: y = 2xy = 2x22

Find the Find the VertexVertex Find Find pp Find the Find the focusfocus Find the Find the directrixdirectrix Make a table showing 5 points Make a table showing 5 points


For the parabola with the equation:For the parabola with the equation: y = 2xy = 2x22 Points:Points: VertexVertex (0,0) (0,0) x y x y pp = = 11//8 8 because because 11//(4p)(4p) = 2 = 2 -2 8 -2 8

focusfocus (0, (0,11//88)) -1 2 -1 2

directrixdirectrix y = - y = -11//88 0 0 0 0

1 21 2

2 2 88


For the parabola with the equation:For the parabola with the equation: y = 2xy = 2x22

VertexVertex (0,0) (0,0) pp = = 11//8 8

focusfocus (0, (0,11//88))

directrixdirectrix y = - y = -11//88

CircleCircle

Definition: Set of all points equidistant Definition: Set of all points equidistant from a given point (from a given point (centercenter). The distance ). The distance is called the is called the radiusradius..

rr

CircleCircle

Graphing the circle: Graphing the circle: The simple equationThe simple equation

is xis x22 + y + y22 = r = r22

The center forThe center for r r this circle is (0,0)this circle is (0,0) r r r r

and its radius is rand its radius is r r r

CircleCircle

Graphing the circle: Graphing the circle: Given the equation : xGiven the equation : x22 + y + y22 = 16 = 16 Give the Give the centercenter Give the Give the radiusradius Give 4 pointsGive 4 points GraphGraph

CircleCircle

Graphing the circle: Graphing the circle: Given the equation : xGiven the equation : x22 + y + y22 = 16 = 16 center center (0,0)(0,0) (0,4)(0,4) radiusradius 44 Give Give 4 points4 points

(4,0), (-4,0)(4,0), (-4,0) (-4,0) (4,0) (-4,0) (4,0)

(0,4), (0,-4)(0,4), (0,-4)

(0,-4)(0,-4)

EllipseEllipse Definition: The set of all points, so Definition: The set of all points, so

that the that the sum of the distancessum of the distances of of each point from 2 given points is each point from 2 given points is constantconstant

The 2 given pointsThe 2 given points

are called are called focifoci

EllipseEllipse

Graphing the EllipseGraphing the Ellipse

The simple equation is:The simple equation is:

x2 y2

a2 b2

EllipseEllipse


In the equation, In the equation, aa is the is the horizontalhorizontal

distance the distance the

ellipse is ellipse is

from the from the

centercenter

EllipseEllipse


In the equation, In the equation, bb is the is the verticalvertical

distance the distance the

ellipse is ellipse is

from the from the

centercenter

EllipseEllipse


The foci The foci cc are on the longest axis of are on the longest axis of

the ellipse.the ellipse.

To find c, To find c,

cc22 = a = a22 – b – b22

or cor c22 = b = b22 – a – a22

EllipseEllipse The ellipse with the equation The ellipse with the equation

xx22 yy22

25 925 9

has the centerhas the center

at (0,0)at (0,0)


xx22 yy22

25 925 9

has a has a horizontalhorizontal - -5 55 5

distance of 5 eachdistance of 5 each

way from the centerway from the center


xx22 yy22

25 925 9

has a has a vertical vertical 33

distance of 3 each distance of 3 each -3-3

way from the centerway from the center


xx22 yy22

25 925 9

has the foci athas the foci at

(-4,0) and (4,0) (-4,0) and (4,0) -4 4-4 4

because cbecause c22 = 25 – 9 = 25 – 9

so c = 4so c = 4

EllipseEllipse If If aa22 is larger is larger, the ellipse is a , the ellipse is a

horizontal ellipse and the foci are on horizontal ellipse and the foci are on the horizontal axisthe horizontal axis

EllipseEllipse If If bb22 is largeris larger, the ellipse is a vertical , the ellipse is a vertical

ellipse and the ellipse and the

foci are on thefoci are on the

vertical axisvertical axis

EllipseEllipse The longest axis is called the The longest axis is called the Major Major

AxisAxis The shortest axis is called the The shortest axis is called the Minor Minor

AxisAxis

Ellipse ExampleEllipse Example

Graphing the EllipseGraphing the Ellipse Given the equation: Given the equation:

xx22 yy22

25 925 9

Give the Give the CenterCenter

Give the Give the VerticesVertices

Give the Give the Co-VerticesCo-Vertices

Give the Give the FociFoci


Graphing the EllipseGraphing the Ellipse Given the equation: Given the equation: xx22 yy22

25 925 9 CenterCenter (0,0) (0,0) Vertices Vertices (on the longest axis) (5,0) & (-5,0)(on the longest axis) (5,0) & (-5,0) Co-Vertices Co-Vertices (on the shortest axis) (on the shortest axis)

(0,3) & (0,-3)(0,3) & (0,-3)


Graphing the EllipseGraphing the Ellipse Given the equation: Given the equation:

xx22 yy22

25 925 9

Foci Foci would be would be cc where c where c22 = a = a22 – b – b22

cc22 = 25 – 9 = 16, so = 25 – 9 = 16, so c = 4c = 4


Graphing the EllipseGraphing the Ellipse Given the equation: Given the equation: (0,3)(0,3)

xx22 yy2 2

25 9 25 9 (-5,0) (5,0)(-5,0) (5,0)

CenterCenter (0,0) (0,0) (-4,0) (4,0) (-4,0) (4,0)

Vertices Vertices (5,0) & (-5,0) (5,0) & (-5,0) (0,-3)(0,-3)

Co-VerticesCo-Vertices (0,3) & (0,-3)(0,3) & (0,-3) FociFoci (4,0), (-4,0)(4,0), (-4,0)

HyperbolaHyperbola Definition: The set of all points so Definition: The set of all points so

that that the differencethe difference of the distances of the distances of the points from 2 given points is of the points from 2 given points is constant.constant.

The 2 givenThe 2 given

points are points are

called called focifoci..

HyperbolaHyperbola Graphing the Hyperbola:Graphing the Hyperbola: The simple equation isThe simple equation is xx22 yy22

aa22 b b22

oror

yy22 xx22

bb22 a a22

HyperbolaHyperbola Graphing the Hyperbola:Graphing the Hyperbola: If the equation has If the equation has xx22 positive, then positive, then xx22 yy22

aa22 b b22

and theand the

hyperbolahyperbola

is is horizontal horizontal

HyperbolaHyperbola Graphing the Hyperbola:Graphing the Hyperbola: If the equation has If the equation has yy22 positive, then positive, then yy22 xx22

bb22 a a22

and theand the

hyperbolahyperbola

is is verticalvertical

HyperbolaHyperbola If the equation isIf the equation is xx22 yy22

aa22 b b22

then thethen the

horizontalhorizontal

hyperbola hyperbola

has vertices of (-a,0) and (a,0) has vertices of (-a,0) and (a,0)

HyperbolaHyperbola If the equation isIf the equation is xx22 yy22

aa22 b b22

then the then the focifoci

are on theare on the

horizontal axis horizontal axis fartherfarther from the origin from the origin than the verticesthan the vertices

HyperbolaHyperbola Graphing the Hyperbola:Graphing the Hyperbola: If the equation isIf the equation is yy22 xx22

bb22 a a22

then thethen the

vertices arevertices are

(0,b) and (0,-b)(0,b) and (0,-b)

HyperbolaHyperbola

To find the value of To find the value of cc, for the , for the foci:foci:

cc22 = a = a22 + b + b22

HyperbolaHyperbola Graphing the Hyperbola:Graphing the Hyperbola: If the simple equation isIf the simple equation is xx22 yy22 Then there are 2Then there are 2

aa22 b b22 lines that tell howlines that tell how

oror wide the hyperbolawide the hyperbola

yy22 xx22 curves will be. curves will be.

bb22 a a22 They are calledThey are called

asymptotesasymptotes..

HyperbolaHyperbola

AsymptotesAsymptotes- are - are lines the curve lines the curve gets closer and gets closer and closer to but closer to but never touchesnever touches

HyperbolaHyperbola Graphing the Hyperbola:Graphing the Hyperbola: If the simple equation isIf the simple equation is xx22 yy22 The equation ofThe equation of

aa22 b b22 the asymptotes forthe asymptotes for

oror either equation iseither equation is

yy22 xx22 y = y = ++ bb//aax x

bb22 a a22

HyperbolaHyperbola Graphing the Hyperbola:Graphing the Hyperbola: Given the equation:Given the equation:

xx22 yy22

16 9 16 9

FindFind the vertices the vertices

Find the Find the focifoci

Give the equation of the Give the equation of the asymptotesasymptotes


xx22 yy22 The The verticesvertices are are

16 9 16 9 (4,0) & (-4,0)(4,0) & (-4,0)

because the xbecause the x22 term term is is positivepositive and and

aa22 = 16, so a = 4 = 16, so a = 4


xx22 yy22 Since bSince b22 = 9, b = 3 = 9, b = 3

16 9 16 9 The The focifoci are found are found by finding by finding cc, which is c, which is c22 = a = a22 + b + b22, or , or cc22 = 4 = 422 + 3 + 322; which means ; which means c = 5c = 5, , the foci are (-5,0) and (5,0)the foci are (-5,0) and (5,0)


xx22 yy22 The equation of the The equation of the

16 9 16 9 asymptotes isasymptotes is

y = y = ++ bb//aaxx

Since a = 4 and b = 3, the equationSince a = 4 and b = 3, the equation

of the asymptotes is y = of the asymptotes is y = ++ 33//44xx


xx22 yy22

16 9 16 9

verticesvertices (4,0) & (-4,0) (4,0) & (-4,0)

focifoci (5,0) & (-5,0) (5,0) & (-5,0)

asymptotesasymptotes y = y = ++33//44x x

conic sections curves with second degree equations

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