11.1 Intro to Conic Sections & The Circle

What is a “Conic Section”?A curve formed by the intersection of a plane and a double rightcircular cone.

We will stu

dy

these in detail

one at a time!

Circles : set of all points in a plane at a fixed distance from a fixed point (radius) (center)

C(h, k)

P(x, y)

Center C(h, k)Any point on circle P(x, y)

By distance formula:2 2

2 2 2

( ) ( )

( ) ( )

r x h y k

r x h y k

r

standard form of a circle

Check out the problems around the room. Work together and answer them all!

1) Find center & radius. x2 + y2 + 8x – 10y = 23

C(–4, 5) r = 8

2) Determine an equation of a circle congruent to the graph of x2 + y2 = 16 and translated 3 units right and 1 unit down.

(x – 3)2 + (y + 1)2 = 16

3) The general form of a circle is x2 + y2 + Dx + Ey + F = 0. *In completing the square if r > 0 circle

r = 0 degenerate circle / point circle

r < 1 the empty set (not possible)Determine what 3x2 + 3y2 – 30x + 18y + 178 = 0 represents.

empty set

4) Determine the equation of the circle that passes through these three points: (5, 3), (–1, 9), (3, –3).

*Use x2 + y2 + Dx + Ey +F = 0 here’s a hint … for (5, 3): 25 + 9 + 5D + 3E + F = 0

x2 + y2 + 4x – 4y – 42 = 0 (x + 2)2 + (y – 2)2 = 50

5) Determine an equation of a circle that satisfies the center at (2, 3) tangent to line 5x + 6y = 14.

*remember! Distance from a point to a line

(x1, y1)

d

Ax + By + C = 0

1 1

2 2

2 2 196( 2) ( 3)

61

Ax By Cd

A B

x y

Homework

#1101 Pg 538 #5, 7, 15, 21, 22, 24–26, 30–32, 34, 36, 38, 41, 45, 47, 49, 51