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