Unit II

References: Chapter3&4: Analytic Geometry by G. Fuller

Chapter11: Analytic Geometry in Calculus by J. Wiley Chapter7: College Algebra by R. D. Gustafson

Conic Sections each one is the intersection of a plane and a right-circular

cone equations fall into one of several categories: a point, a pair

of lines, a circle, a parabola, an ellipse, an hyperbola, or no graph at all

Ren Descartes (15961650) and Blaise Pascal (16231662) developed the mathematics needed to study them in detail

Unit II : Conics

Definition:

A circle is the set of all points on a plane that are equidistant from a fixed-point on the plane. The fixed point is called the center. And the distance from the center to any point of the circle is called radius.

The Standard Equation of a Circle with Center at (h, k) :

The Standard Equation of a Circle with Center at (0, 0):

222 rkyhx

222 ryx

Derive the general form of the equation of a circle in second degree in x and y : Standard equation of a circle with center (h,k)

Equation can be written as

General form of the equation of a circle in second degree in x and y

022 FEyDxCyBxyAx

022 FEyDxyx

222 rkyhx Square the binomials and simplify

where B=0, A=C and D, E and F are constants

Find the general equation of the circle with center at (-3, 4) and radius =6 units.

Find the center and radius and sketch the circle.

Find the equation of the circle which is concentric with the circle and that goes through the point (-1,3).

Find the equation of the circle determined by the given conditions; draw the figure. Tangent to the line , Center at (5,5)

0154622 yxyx

0510822 yxyx

01034 yx

Find the equation of the circle with center at (-5,6) and radius = 4.

Find the equation of the circle with diameter from (4,2) to (8,6).

Find the center, the radius and sketch the ff. circles:a)

b)

Find the equation of the circle whose center is at (3,4) and which is tangent to the y-axis.

08622 yxyx

025322 22 yxyx

Unit II: Conics

Definition: A parabola is the set of all points

in a plane equidistant from a line l(called the directrix) and a fixedpoint F (called focus) that is not online l.

The point on the parabola that isclosest to the directrix is called thevertex, and the line passingthrough the vertex and the focus iscalled the axis of the parabola.

Using the distance formula, find the distance from P(x,y)to F(0,p).

From the figure, we see that the distance from P() to the directrix y=-p is:

A parabola with the following characteristics: Vertex: V(0,0) Focus: F(0,p) Directrix: y= -p Point on the parabola: P(x,y)

22)( pyxPFd

pypy

We can equate these distances and simplify :

Equation (1), one of the standard equations of a parabola with vertex at the origin.

If p>0 in equation(1), the graph of the equation will be a parabola that opens upward. If p

Standard Form of the Equation of a Parabola with Vertex (0, 0)

Equation 2 = 4 2 = 4 Focus 0, , 0 Directrix = = Axis of Symmetry Vertical, y-axis Horizontal, x-axis

Latus Rectum 2 2 Length of the Latus Rectum

4 4

p>0 opens up opens right

p

x2 = 4py opens up

x2 = 4py opens down

y2 = 4px opens right

y2 = 4px opens left

Standard Form of the Equation of a Parabola with Vertex (h, k)

Equation 2 = 4 2 = 4 Focus , + + ,

Directrix = =

Axis of Symmetry Vertical, y-axis Horizontal, x-axis

Latus Rectum 2 2

Length of the Latus Rectum

4 4

p>0 opens up opens right

p

(x-h)2 = 4p(y-k) opens up

(x-h)2 = 4p(y-k) opens down

(y-k)2 = 4p(x-h) opens right

(y-k)2 = 4p(x-h) opens left

General form of the equation of a parabola:

02 FEyDxx

02 FEyDxy

Provided E0

Provided D0

Write the equation of the parabola with vertex at origin and focus at (0,4).

A parabola has its vertex at the origin, its along the x-axis, and passes through the point (-3,6). Find the equation.

Draw the graph of the equation

025682 yxy

Find the equation of the parabola that opens up, has vertex at the point (4,5), and passes through the point (0,7).

Find the equations of two parabolas with a vertex (2,4) at that pass through (0,0).

Find the vertex and y-intercepts of the parabola with the following equation and graph it: 28482 yxy

1. Find and graph the vertex, focus, and directrix of each parabola.

a)

b)

2. Find and graph the equation of each parabola.

a) Vertex at (0,0); focus at (0,3)

b) Vertex at (1,-5); directrix at x=-1

c) Vertex at (6,8); passes through (5,10) and (5,6)

yx 122

xy 203 2

Unit II: Conics

Definition: An ellipse is the set of all points P in a plane such

that the sum of the distances from P to two other fixed point F and F is a positive constant. Each of the fixed points is called focus (plural foci)

Derive the equation of the ellipse: By defintion of an ellipse:

Use the distance formula to compute the lengths of FP and PF.

Square both sides of equation(2) and simplify:

Square both sides and simplify:

The shortest distance between two points is a line segment, d(FP) + d(PF) > d(FF)

Therefore, 2a > 2c thus, a > c and a2 - c2 is positive number which will be called b2.

Let b2 = a2 - c2, substitute to equation(3)

Dividing both sides of equation(4) by a2b2 gives the equation:

To find the coordinates of the vertices V and V, we substitute 0 for y and solve for x:

The coordinates of V are (a,0) and the coordinates of V are (-a,0), a is the distance between the center of the ellipse and either of its vertices.

The center of the ellipse is the midpoint of the major axis.

To find the coordinates B and B, we substitute 0 for x and solve for y:

The coordinates of B are (0,b) and the coordinates of B are (0,-b) , the distance between the center of the ellipse and either endpoint of the minor axis is b.

Major Axis on x-axis, Center (0,0)

Major Axis on y-axis, Center (0,0)

Major Axis Horizontal, Center (h,k)

Major Axis Vertical, Center (h,k)

Note:

The larger denominator in the equation of an ellipse determine whether the foci, the vertices, and the major axis are along the x or y axis.

where a>c

Eccentricity, , e

General form of the equation of a ellipse:

022 FEyDxCyAx

Provided AC

Sketch the following: (Label all the input parts)

a)

b)

Find the equation of the ellipse for the ffcondition: (Sketch)

a) Vertices at (5, 0) and foci at (4,0)

b) Vertices at (0, 10), eccentricity

0225259 22 yx

015464916 22 yxyx

5

4

Find the equation of the ellipse for the ffcondition: (Sketch)

a) Minor axis length = 12 units, distance between foci = 16, center at origin and x-axis as the major axis.

b) Eccentricity, e = , foci on the y-axis, center at the origin and passing through (6,4)

Find the equation of the graph of all points the sum of whose distances from (3,0) and (9,0) is 12 units.

Unit II: Conics

Definition: An hyperbola is the set of all points P in a plane

such that the difference of the distances of each point of the set from two fixed points(foci) in the plane is constant.

Terms: Foci of the hyperbola

points F and F

Center the midpoint of chord FF

Vertices points V and V, where the

hyperbola intersects FF

Transverse Axis segment VV

Conjugate Axis segment BB

Foci on x-axis, Center at (0,0)

Foci on y-axis, Center at (0,0)

Transverse Axis Horizontal, Center at (h,k)

Transverse Axis Vertical, Center at (h,k)

Fundamental Rectangle

Form rectangle RSPQ

Asymptotes of the Hyperbola

The extended diagonals of the rectangle

Equations of the Asymptotes

Foci on x-axis, Center at (0,0)

and

Foci on y-axis, Center at (0,0)

and

xa

by x

a

by

xb

ay x

b

ay

Equations of the Asymptotes

Foci on x-axis, Center at (h,k)

and

Foci on y-axis, Center at (h,k)

and

a

bhkx

a

by

a

bhkx

a

by

b

ahkx

b

ay

b

ahkx

b

ay

General form of the equation of a hyperbola:

022 FEyDxCyAx

Graph the hyperbola: a)

Write the equation of the hyperbola:a) vertices V(4,0) and V(-4,0) and a focus at F(5,0)

b) vertices (3,-3) and (3, 3) and a focus at (3,5)

Find the area of the fundamental rectangle of the hyperbola:a)

124222 yxyx

66422 yxyx

Unit II: Conics

Simplification

Examples By translation of axes simplify the equation:

Given the equation of a function

, translate the origin so that the new equation will have no first-degree terms.

015662 yxx

0108201652 22 yxyx

hxx ' kyy '

cos'sin'

sin'cos'

yxy

yxx

RPNSRPMRMPy

RSONMNONOMx

Rotation Formulas:

Example:

Transform the equation , by rotating the axes through 45.

Find the angle of rotation such that the transformed equation will have no xy term.

0922 yx

023 yxy

Unit II: Conics

Fin dthe longest and shortest distance from (10,7) to the circle

The orbit of the earth is an ellipse with the sun at a focus, the semi-major 93 million miles, and the eccentricity is 1/60. Find the greatest and least distance of the earth from the sun.

0202422 yxyx

An arch in the form of a parabolic curve with a vertical axis is 60 m across the bottom. The highest point is 16 m above the horizontal base. What is the length of a beam placed horizontally across the arch 3 m below the top?

An overpass arc is semi-elliptical, 40 m across the base and 15 m high at the center. What is the maximum height of truck, 10 m wide at the top which can pass through the overpass.