Conic Sections Formulas

Circle

Circle

(x-h)²+(y-k)²=r²

Area of a Circle

πr²

Degenerate Circle

(x-h)²+(y-k)²=0

How can you tell if an equation is a circle without completing the square?

• both variables are squared, somewhere in the equation

• both x² and y² have exactly the same coefficient and the same sign

Parabola (up/down)

y= a(x - h)² +k

Parabola (left/right)

x= a(y - k)² +h

Parabola (a)

a = 1/4c

Parabola (c)

c = 1/4a

Length of Latus Rectum (parabola)

I 1/a I = I 4c I

C= dist. from . . .

vertex to focus or from vertex to directrix

How can you tell if an equation is a parabola without completing the square?

• only x or y is squared, but not both

• if x is squared, then the graph goes up (a>0) or down (a<0)

• if y is squared, then the graph goes right (a>0) or left (a<0)

Ellipses formula (stretched horizontally)

(x - h)² + (y - k)² = 1

a² b²

Ellipses formula (stretched vertically)

(y - k)² + (x - h)² = 1

a² b²

Area of interior of ellipse

πab

Sum of focal radii

2a

length of major axis (ellipses)

2a

length of minor axis

2b

C= distance from . . .

center to each of the foci

a²= (ellipses)

b² + c²

How do you know if an equation is an ellipse without completing the square?

• both of the variables are squared, somewhere in the equation

• coefficients of x² and y² are different

• signs of the coefficients of x² and y² are the same

• the square root of the number under the “x” term will indicate the horizontal stretch

• the square root of the number under the “y” term will indicate the vertical stretch

Hyperbola formula (left/right)

[(x - h)²/ a²] - [(y - k)²/ b²] = 1

hyperbola formula (up/down)

[(y - k)²/ a²] - [(x - h)²/ b²] = 1

Difference of focal radii

2a

c= distance from . . . . (hyperbola)

center to each focus

c²=

a² + b²

How do you know if an equation is a hyperbola without completing the square?

• both variables are squared, somewhere in the equation

• coefficients of x² and y² may be the same or may be different

• signs of coefficients of x² and y² are different

• if the x term is positive/first, then the graph is “stretched” horizontally, if the y term is positive/first then the graph is “stretched” vertically

Degenerate Hyperbola

[(x - h)²/ a²] - [(y - k)²/ b²] = 0

Degenerate Ellipse

[(x - h)²/ a²] + [(y - k)²/ b²] = 0