10.1Conics and Calculus

Each conic section (or simply conic) can be described as the intersection of a plane and a double-napped cone.

Circle Parabola Ellipse Hyperbola

Conic Sections

General second-degree equation

Each of the conics can be defined as a collection of points satisfying a certain geometric property,

2 2 0Ax Bxy Cy Dx Ey F

A circle can be defined as the collection of all points (x, y) that are equidistant from a fixed point (h, k). This definition easily produces the standard equation of a circle

Circle

2 2 2( ) ( )x h y k r

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

The midpoint between the focus and the directrix is the vertex, and the line passing through thefocus and the vertex is the axisof the parabola.

The standard form of the equation of

a parabola with vertex (h,k) is

Parabolas

2

2

( ) 4 ( )

( ) 4 ( )

x h p y k

y k p x h

Vertical axis

Horizontal axis

Find the focus of the parabola given by

Example21 1

2 2y x x

22 1 2y x x

2 2 1 2x x y 2 2 1 2 2x x y

2( 1) 2( 1)x y

An ellipse is the set of all points (x, y) the sum of whose distances from two distinct fixed points called foci is constant.

The line through the foci intersects the ellipse at two points, called the vertices.

Ellipses

The chord joining the vertices is the major axis, and its midpoint is the center of the ellipse.

The chord perpendicular to the major axis at the center is the minor axis of the ellipse.

The standard form of an ellipse with the center (h, k) and major and minor axes of length 2a and 2b, where a > b, is

The foci lie on the major axis, c units from the center, with

Ellipses

2 2

2 2

2 2

2 2

( ) ( )1

( ) ( )1

x h y k

a bor

x h y k

b a

2 2 2c a b

Horizontal major axis

Vertical major axis

ExampleFind the center, vertices and the foci of the ellipse given by

2 2

116 9

x y

Find the center, vertices, and foci of the ellipse given by

4x2 + y2 – 8x + 4y – 8 = 0.

Solution: 4x2 – 8x + y2 + 4y = 8

4(x2 – 2x + 1) + (y2 + 4y + 4) = 8 + 4 + 4

4(x – 1)2 + (y + 2)2 = 16

Example

2 2( 1) ( 2)1

4 16

x y

h = 1, k = –2, a = 4, b = 2, and

c = Center: (1, –2)

Vertices: (1, –6) and (1, 2)

16 4 12

To measure the ovalness of an ellipse, we define the eccentricity e to be the ratio

Eccentricity

ce

a

Notice that 0 < c < a, and thus 0 < e < 1.

For an ellipse that is nearly circular, e is very small, and

for an elongated ellipse, e is close to 1.

If e > 1, we will have a hyperbola.

A hyperbola is the set of all points (x, y) for which the absolute value of the difference between the distances from two distinct fixed points called foci is constant.

The line through the two fociintersects a hyperbola at two pointscalled the vertices.

The line segment connecting the vertices is the transverse axis, and the midpoint of the transverse axis is the center of the hyperbola.

A hyperbola has two separate branches, and has two asymptotes that intersect at the center.

Hyperbolas

The standard form of a hyperbola with center (h, k) and transverse axis of length 2a is

The foci lie on the transverse axis, c units from the center, with

Hyperbola

2 2

2 2

2 2

2 2

( ) ( )1

( ) ( )1

x h y k

a bor

y k x h

a b

2 2 2c a b

Horizontal Transverse Axis

Vertical Transverse Axis

The asymptotes pass through the vertices of a rectangle of dimensions 2a by 2b, with its center at (h, k).

Sketch the graph of the hyperbola whose equation is

4x2 – y2 = 16.

Solution:

h = 0, k = 0, a = 2, b = 4 , c =

The transverse axis is horizontal and the vertices occur at (–2, 0) and (2, 0).

The ends of the conjugate axis occur at (0, –4) and (0, 4).

Example

2 2

14 16

x y

4 16 20