4.4 Conics

• Recognize the equations and graph the four basic conics: parabolas, circles, ellipse, and hyperbolas.

• Write the equation and find the focus of a

parabola.

• Write the equation of a ellipse and find the

foci, vertices, the length of the major and minor

axis.

Parabolas

Definition: A parabola is the set of all points (x, y) in a plane that are equidistant from a fixed line, called the directrix, and a fixed point, the focus, not on the line.

Directrix

Parabolas

x2 = 4py p 0Vertex (0, 0)Directrix y = -pFocus (0, p)Line of sym x = 0

y2 = 4px p 0Vertex (0,0)Directrix x = -pFocus (p, 0)Line of sym y = 0

Standard equation of the Parabola

Parabola ExamplesGiven 22xy Find the focus. Since the squared variable is x, the

parabola is oriented in the y directions. The leading coefficient is negative, so, the parabola opens down.

-pp

Focus (0, p)

•

Solve for x2

pyyx

yx

42

12

1

2

2

Solve for p.

p

p

8

1

42

1

Focus )8

1,0(

Parabola ExamplesWrite the standard form of the equation of the parabola with the vertex at the origin and the focus (2, 0).

Note that the focus is along the x axis, so the parabola is oriented in the x axis direction, y2 = 4px.

Focus (2, 0) (p, 0)

p = 2

y2 = 4px = 4(2)x

y2 = 8x

Classwork

Page 370 problems 9 –14.Page 371 problems 17 – 28.

Ellipse

Definition: An ellipse is the set of all points (x, y ) in a plane the sum of whose distances from two distinct fixed points (foci) is constant.

• •

•

Focus Focus

(x, y)

d1 d2

d1 + d2 = constant

Vertex Vertex

MajorAxis

MinorAxis

•

•

•

• •Center

Ellipse

The standard form of the equation of an ellipse (Center at origin)

1or 12

2

2

2

2

2

2

2

a

y

b

x

b

y

a

x

where 0 < b < a c2 = a2 – b2

Major axis length = 2a Minor axis length = 2b

Major axis along the x axis

Major axis along the y axis

Ellipse Examples

Given 4x2 + y2 = 36. Find the vertices, the end points of the minor axis, the foci and center.

Change the equation to the standard form.

1369

36

36

3636

4

22

2

yx

yx

a2b2

Major axis along the y-axis

Center (0, 0) Vertices (0, 6) End points of minor axis (3, 0)

To find the foci use c2 = a2 – b2

c2 = 36 – 9 c2 = 27

)33,0( Foci

33

c

(0, a) (b, 0)

(0, c)

Classwork

Page 372 problems 35 – 40 45 – 55

Hyperbolas

Definition: a hyperbola is the set of all points (x, y) the difference of whose distances from two distinct points (foci) is constant.

••

•

(x, y) ••

•

d1

d2

d1 - d2 = constant

••

•Center

Vertex

Vertex

Focus

Focus

Transversal Axis

Branch

Branch