Conic Sections

ParabolaWill have only one variable squared

• y=+x2 x=+y2

• y=-x2 x=- y2

Parabola

• Non squared term needs to be by itself and postive

x=a(y-k)2+h y=a(x-h)2+k

• Vertex for both forms (h, k)

• May need to complete the square to get into this form

CircleWill have both variables squared,

added and with the same coefficients

• (x-h)2 + (y-k)2=r2

• Center (h, k)• Radius r (h,k)

• (x-2)2 + (y+3)2=9 .____

• Center (2, -3) radius 3 r

Ellipse Will have both variables squared, added and with different coefficients. Always equals 1.

1)()(

2

2

2

2

b

ky

a

hx

a2 > b2

b2 > a2

Center = (h, k)

Move right and left of center point

Move up and down of center point

2a2b

Ellipse• If not equal to 1 divide entire equation by constant

• May need to complete the square to get into standard form (could be x, y or both)

14

)2(

3

)3(

12

12

12

)2(3

12

)3(4

12)2(3)3(4

22

22

22

yx

yx

yx

Hyperbola Will have both variables squared and

subtracted. Always equals 1.

1)()(

1)()(

2

2

2

2

2

2

2

2

a

hx

b

ky

b

ky

a

hx

a

b

Asymptote sloperun

risechangex

changey

1)()(

1)()(

2

2

2

2

2

2

2

2

b

hx

a

ky

b

ky

a

hx

Hyperbola

• If not equal to one divide by constant (refer to ellipse example)

• May need to complete the square to get into standard form (could be x, y or both)

What conic will the equation create?

258.

01004.

125.

02512.

4)1(.

22

22

22

2

22

yxxE

yxD

yxyC

xyyB

xyA

129.

25)3(.

48416.

070710.

6424425.

2

22

22

22

22

xxyJ

yxI

yxH

yxG

yyxF