MA – 9.1 & 9.2 – Conics Intro, Circles, Parabolas

Types of Conics

Circles

·  We find the radius using the distance formula. r=x2-x12+y2-y12

·  Standard form of a circle is x-h2+y-k2=r2 for center (h,k) and radius, r.

·  General form of a circle is x2+y2+ax+by+c=0.

Example: Identify the center and radius of each circle.

a)  x-22+y-42=36

b)  x+52+y2=25

c)  x2+y-22=19

Example: Identify the center and radius of each circle by writing the equation in standard form:

x2+y2-8x+6y-11=0.

Example: Identify the center and radius of a circle with the equation x2+y2+y=34.

Parabolas

A parabola is all points that are the same distance away from a fixed point, F, as they are from a fixed line D. We call this fixed point the focus and the fixed line the directrix. The line that goes through the focus and directrix is called the axis of symmetry and the intersection of the parabola with the axis of symmetry is called the vertex.

The general form of a parabola is different depending on the axis of symmetry.

Vertical Axis / Horizontal Axis
General Form Equation / x-h2=4py-k / y-k2=4px-h
Axis of Symmetry / x=h / y=k
Vertex / h,k / h,k
Focus / h,k+p / h+p,k
Focal Chord (Latus Rectum) / /
Directrix / y=k-p / x=h-p
Direction of Opening / Up if p>0
Down if p<0 / Right if p>0
Left if p<0

*p is the distance between the vertex and the directrix or the vertex and the focus and a≠0

Example: Find the vertex, focus, and directrix of each parabola.

·  x2=8y **Graph this!

·  y2=4x **Graph this!

·  x+42=16(y+2)

·  y2-4y+4x+4=0

·  x2+8x=4y-8

Example: Write the equation of the parabola described.

·  Directrix at x=1, Focus at (-3,-2)

·  Vertex at (4,-2), Focus at (6,-2)

·  Vertex at (-1,4), Directrix at y=2