Michael FarellaConics Memory Aid

Math SN5May 25, 2013

Circles Locus definition of a circle: The locus of

points a given distance from a given point in that plane.

Rule for a circle: (x-h)2+(y-k)2=rWhere h is the horizontal translationand k is the vertical translation. R isThe radius of the circle. The only Properties of a circle are the domainAnd range which are both equal to2r.

EllipseLocus definition of an ellipse: the locus of

points whose distances to a fixed point and a fixed line are in a constant ratio less than 1.

Rule: (x-h)2/a2+(y-k)2/b2=1Where h and k are the horizontalAnd vertical translation. A Represents the semi major axisAnd be represents the semi minor axis. When a

is greater than b the oval is horizontal and when b is greater than a the oval is vertical. If ellipse is vertical the rule is (x-h)2/b2+(y+k)2/a2=1

EllipseF1 and F2 represent the foci of theEllipse, if the oval is horizontalThe foci will be on the x axis and If it is vertical, they are on the y Axis. The foci is represented by c And to find the foci we use c2=a2-b2.

Ellipse

Ellipse

Ellipse The focal radii always add up to a constant

value in any given ellipse. L1 + l2 will Always come to the same sum.

HyperbolaLocus definition: The difference of whose

distances from two fixed points is a constant.

Rule: x2/a2-y2/b2=1 Rule: y2/b2-x2/a2=1

HyperbolaRule of a hyperbola: (x-h)2/a2-(y+k)2/b2

H and k are the horizontal and vertical translation. A is the distance between the hyperbola and the center on the x axis. B is the distance between the rectangle and the center. If it is a vertical hyperbola the roles of a and b switch.

HyperbolaTo find the foci of a hyperbola it is c2=a2+b2

where c is the foci. For example...A=2 and b=4C2=22+42 C2=20C=4.47So the foci for this hyperbolaIs (+-4.47,0)Finding the asymptotes:Asymptotes= -+(b/a)=-+(4/2)=2xAsymptotes= y=-+2x

Hyperbola

ParabolaLocus definition: the locus of a point that

moves so that it is always the same distance between the focus and the directrix.

Rule: y=a(x-h)2+k rule: y=-a(x-h)2+k

H and k affect the horizontal and vertical translation.

ParabolaOther parabolas:Rule: x=a(y-k)2+h Rule: x=-a(y-

k)2+h

Parabola

ParabolaNo matter which point on the parabola, the

distance between the focus and the point and the directrix and the point will always be the same.

ParabolaThe distance between the vertex and the

directrix and the vertex and the focus will always be the same and this value is represented by c. We need to find c to find the coordinates of the focus. Focus=(0,c).

To find c we use the formulaC=1/4a.

ParabolaParabolas are formed by connecting two points

on two lines.

A segment of a parabola is called a lissajous curve which is the graph of a system of parametric equations.

Different forms of rules

The parabola can be expressed with 2 other forms:

F(x)=ax2+bx+c or f(x)=a(x-x1)(x-x2)

Find the ruleFind rule of parabola who’s focus is at (0,5) and

who’s vertex is at the origin of the graph.C=1/4a5=1/4aA=0.05F(x)=0.05x2

Find the ruleFind the rule of a hyperpola with the

asymptotes Y=+-1/4x and the vertices are +-3.5 and it is a

vertical hyperbola.

Find the rule of hyperpolaAs we can see a=3.5 and b=1.So the rule for this hyperbola is x2/12.25-y2/1=1