Conic Sections In Standard Form
	Circles	Parabolas	Ellipses	Hyperbolas
Horizontal:
Equation	(x-h)² + (y-k)² = r²	x = a(y-k)² + h	(x-h)²/a² + (y-k)²/b² = 1	(x-h)²/a² - (y-k)²/b² = 1
Center	(h, k)		(h, k)	(h, k)
Axis Of Symmetry	Any Diameter	y = k	See “a” and “b”	See “a” and “b”
Focus (Foci)	(h, k) center is focus	(h+1/(4a), k)	(h+c, k) & (h-c, k)	(h+c, k) & (h-c, k)
Directrix		x = h - 1/(4a)
Vertex (Vertices)		(h, k)	(h+a, k), (h-a, k), (h, k+b), (h, k-b)	(h+a, k), (h-a, k)
Asymptotes				y= ±(b/a)(x-h)+k
Variables	r = radius		a = semi-major axis	a = semi-transverse axis
			b = semi-minor axis	b = semi-conjugate axis
Note:			a² > b²
"c" is equal to…			sqrt ( a²-b² )	sqrt (a²+b² )
Eccentricity:	e = 0	e = 1	e = c/a; 0<e<1	e = c/a; e>1
Vertical:
Equation	(x-h)² + (y-k)² = r²	y = a(x-h)²+k	(x-h)²/b² + (y-k)²/a² = 1	(y-k)²/a² - (x-h)²/b² = 1
Center	(h, k)		(h, k)	(h, k)
Axis Of Symmetry	Any Diameter	x = h	See “a” and “b”	See “a” and “b”
Focus (Foci)	(h, k) center is focus	(h, k+1/(4a))	(h, k+c) & (h, k-c)	(h, k+c) & (h, k-c)
Directrix		y = k - 1/(4a)
Vertex (Vertices)		(h, k)	(h+b, k), (h-b, k), (h, k+a), (h, k-a)	(h, k + a), (h, k - a)
Asymptotes				y= ±(a/b)(x-h)+k
Variables	r = radius		a = semi-major axis	a = semi- transverse axis
			b = semi-minor axis	b = semi- conjugate axis
Note:			a² > b²
"c" is equal to…			sqrt (a²-b²)	sqrt (a²+b²)
Eccentricity:	e = 0	e = 1	e = c/a; 0<e<1	e = c/a; e>1

For Rectangular Hyperbolas See the Lecture Notes in your packet or on the Web:

http://www.lcusd.net/lchs/dclausen/algebra2/rectangular_hyperbolas.htm

Distance Formula:

Midpoint Formula:

Conic Sections in General Form: Ax² + By² + Cx + Dy + E = 0

Circles:

For Circles A = B

A & B need to have the same number and same sign.

Parabolas:

If either A = 0 or B = 0 then the equation defines a parabola (x² or y² is missing). Isolate the variable that is not squared and use the completing the square method to convert the equation to that of a Parabola in Standard Form.

Hint: Solve for y if there is no y² in the equation or solve for x if there is no x² in the equation.

Ellipses:

1) A & B have the same sign (both positive or both negative)

2) A & B are different numbers (if they were the same, this would be a circle).

Hyperbolas:

1) A & B must have different signs (one positive and the other negative).

2) A & B must be different numbers or opposites (same number with different signs)