Conics Memory Aid Math SN5 May 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 =r Where h is the horizontal translation and k is the vertical translation. R is The radius of the circle. The only Properties of a circle are the domain And range which are both equal to 2r.

Ellipse Locus 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 /a 2 +(y-k) 2 /b 2 =1 Where h and k are the horizontal And vertical translation. A Represents the semi major axis And 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 /b 2 +(y+k) 2 /a 2 =1

Ellipse F1 and F2 represent the foci of the Ellipse, if the oval is horizontal The 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 c 2 =a 2 -b 2.

Ellipse

The focal radii always add up to a constant value in any given ellipse. L1 + l2 will Always come to the same sum.

Hyperbola Locus definition: The difference of whose distances from two fixed points is a constant. Rule: x 2 /a 2 -y 2 /b 2 =1 Rule: y 2 /b 2 -x 2 /a 2 =1

Hyperbola Rule of a hyperbola: (x-h) 2 /a 2 -(y+k) 2 /b 2 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.

Hyperbola To find the foci of a hyperbola it is c 2 =a 2 +b 2 where c is the foci. For example... A=2 and b=4 C 2 = C 2 =20 C=4.47 So the foci for this hyperbola Is (+-4.47,0) Finding the asymptotes: Asymptotes= -+(b/a) =-+(4/2)=2x Asymptotes= y=-+2x

Hyperbola

Parabola Locus 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.

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

Parabola

No 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.

Parabola The 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 formula C=1/4a.

Parabola Parabolas 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)=ax 2 +bx+c or f(x)=a(x-x1)(x-x2)

Find the rule Find rule of parabola who’s focus is at (0,5) and who’s vertex is at the origin of the graph. C=1/4a 5=1/4a A=0.05 F(x)=0.05x 2

Find the rule Find the rule of a hyperpola with the asymptotes Y=+-1/4x and the vertices are and it is a vertical hyperbola.

Find the rule of hyperpola As we can see a=3.5 and b=1. So the rule for this hyperbola is x 2 /12.25-y 2 /1=1