Conic Sections and Other Cool Mathematicalness

That was the title of our class today.
Today was our introduction to conics and we learned there are 4 sections to conics.

1. circle
2. ellipse
3. hyperbola
4. parabola

This is a conic: x^2 +/- y^2
General form of a conic is: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
*A, B, C, D, E and F are elements of the set of reals. B cannot equal 0
Conics are always squared

A circle happens when the A value and the C value are the same.

(x^2 + y^2) = 1 is a small circle
(x^2 + y^2) = 7 is a bigger circle...etc

Ex. Given the general form of the conic equation,

a) Identify the conic
b) State A, C, D, E, F

given... x^2 + y^2 - 8 = 0
a) So this is a circle because both co-efficients are 1.
b) A = 1, C = 1, D = 0, E = 0, F = -8

given... 2x^2 + 2y^2 + 4x -2y -32 = 0
a) This is a circle, because A and C values are the same
b) A = 2, C = 2, D = 4, E = -2, F = -32

An ellipse happens when A and C values are the same sign, but A cannot equal C. An ellipse is a squashed circle, an ellipse is also a subset of a circle.

Ex. Given the general form of the conic equation,
a) Identify the conic.
b) State A, C, D, E, F

given...(1) x^2 + 49y^2 - 49 = 0
a) Ellipse because both A and C are positive, but the values are different.
b) A = 1, C = 49, D = 0, E = 0, F = -49

given... 4x^2 + 9y^2 - 3x + 2y +0
a) Ellipse because both A and C are positive, but the values are different.
b) A = 4, C = 9, D = -3, E = 2, F = 0

A hyperbola happens if A and C have opposite signs.
*If the x is positive, than the hyperbola will open horizontally, but if the x is negative the hyperbola will open vertically.

Ex. Given the general form of the conic equation,
a) Identify the conic
b) State A, C, D, E, F

given... 9x^2 - 4y^2 - 36 = 0
a) Hyperbola because the A value is positive, and the C value is negative.
b) A = 9, C = -4, D = 0, E = 0, F = -36

given... -3x^2 + 3y^2 +2x - 12y + 2 = 0
a) Hyperbola because A is negative and C is positive.
b) A = -3, C = 3, D = 2, E = -12, F = 2

If one of A or C is the value of 0, than you have a parabola.

Ex. Given the general form of the conic equation,
a) Identify the conic
b) State A, C, D, E, F

given... y^2 - 4x = 0
a) Parabola because A value is 0
b) A = 0, C = 1, D = -4, E = 0, F = 0

given... 3x^2 - 2x + 5y - 3 = 0
a) Parabola because A ihas a value bigger than 0, and C's value is 0.
b) A = 3, C = 0, D = -2, E = 5, F = -3

The exercises we were assigned are:
Exercise 36 #1-7
Exercise 37 #1-11
Exercise 38 #1-9

Dont panic because there is no way you wil be able to do all these questions with just the information we learned on conics yesterday. You need more information.