Definition of a circle
 A circle is the set of all points in the plane that
are equidistant from a given point in the plane
called the center. The distance from the center
to any point on the circle is called the radius.
 The standard form of the equation of a circle
is (x-h) + (y-k) = r where (h,k) is the
center of the circle and r is the radius.
2
2
2
A little bit of history
 A circle is one type of conic section. Conic
sections were first studied in ancient Greece
sometime between 600 and 300 BC. The
Greeks were largely concerned with the
properties not the applications of conics. In
the 17th century, applications of conics
became important in the development of
calculus.
Ellipses in real life examples
 Though not so simple as the circle, the ellipse is nevertheless the curve
most often "seen" in everyday life. The reason is that every circle, viewed
obliquely, appears elliptical. Any cylinder sliced on an angle will reveal an
ellipse in cross-section (as seen in the Tycho Brahe Planetarium in
Copenhagen).
Ellipses in your glass!
 Tilt a glass of water and the surface of the
liquid acquires an elliptical outline. Salami is
often cut obliquely to obtain elliptical slices
which are larger.
Space and ellipses
 The early Greek astronomers thought that the planets moved in circular orbits about
an unmoving earth, since the circle is the simplest mathematical curve. In the 17th
century, Johannes Kepler eventually discovered that each planet travels around the
sun in an elliptical orbit with the sun at one of its foci. The orbits of the moon and of
artificial satellites of the earth are also elliptical as are the paths of comets in
permanent orbit around the sun. Halley's Comet takes about 76 years to travel
abound our sun. Edmund Halley saw the comet in 1682 and correctly predicted its
return in 1759. Although he did not live long enough to see his prediction come true,
the comet is named in his honour.
 On a far smaller scale, the electrons of an atom move in an approximately
elliptical orbit with the nucleus at one focus.
Properties of ellipses
 The ellipse has an important property that is used in the reflection
of light and sound waves. Any light or signal that starts at one
focus will be reflected to the other focus. This principle is used in
lithotripsy, a medical procedure for treating kidney stones. The
patient is placed in a elliptical tank of water, with the kidney stone
at one focus. High-energy shock waves generated at the other
focus are concentrated on the stone, pulverizing it.
 The principle is also used in the construction of "whispering
galleries" such as in St. Paul's Cathedral in London. If a person
whispers near one focus, he can be heard at the other focus,
although he cannot be heard at many places in between.
 Statuary Hall in the U.S. Capital building is elliptic. It was in this
room that John Quincy Adams, while a member of the House of
Representatives, discovered this acoustical phenomenon. He
situated his desk at a focal point of the elliptical ceiling, easily
eavesdropping on the private conversations of other House
members located near the other focal point.
Parabolas
 THE PARABOLA One of nature's best known approximations to parabolas
is the path taken by a body projected upward and obliquely to the pull of
gravity, as in the parabolic trajectory of a golf ball. The friction of air and
the pull of gravity will change slightly the projectile's path from that of a
true parabola, but in many cases the error is insignificant. This discovery
by Galileo in the 17th century made it possible for cannoneers to work out
the kind of path a cannonball would travel if it were hurtled through the air
at a specific angle. When a baseball is hit into the air, it follows a parabolic
path; the center of gravity of a leaping porpoise describes a parabola. The
easiest way to visualize the path of a projectile is to observe a waterspout.
Each molecule of water follows the same path and, therefore, reveals a
picture of the curve. The fountains of the Bellagio Hotel in Las Vegas
comprise a parabolic chorus line.
Properties of parabolas

Parabolas exhibit unusual and useful reflective properties. If a light is
placed at the focus of a parabolic mirror (a curved surface formed by
rotating a parabola about its axis), the light will be reflected in rays
parallel to said axis. In this way a straight beam of light is formed. It is
for this reason that parabolic surfaces are used for headlamp reflectors.
The bulb is placed at the focus for the high beam and a little above the
focus for the low beam. The opposite principle is used in the giant
mirrors in reflecting telescopes and in antennas used to collect light and
radio waves from outer space: the beam comes toward the parabolic
surface and is brought into focus at the focal point. The instrument with
the largest single-piece parabolic mirror is the Subaru telescope at the
summit of Mauna Kea in Hawaii (effective diameter: 8.2 m). Heat
waves, as well as light and sound waves, are reflected to the focal point
of a parabolic surface. If a parabolic reflector is turned toward the sun,
flammable material placed at the focus may ignite. (The word "focus"
comes from the Latin and means fireplace.) A solar furnace produces
heat by focusing sunlight by means of a parabolic mirror arrangement.
Light is sent to it by set of moveable mirrors computerized to follow the
sun during the day. Solar cooking involves a similar use of a parabolic
mirror. Two types of images exist in nature: real and virtual. In a real
image, the light rays actually come from the image. In a virtual image,
they appear to come from the reflected image - but do not. For
example, the virtual image of an object in a flat mirror is "inside" the
mirror, but light rays do not emanate from there. Real images can form
"outside" the system, where emerging light rays cross and are caught as in a Mirage, an arrangement of two concave parabolic mirrors.
Mirage's 3-D illusions are similar to the holograms formed by lasers.
Hyperbolas
A
sonic boom shock wave has the shape of a cone, and it intersects
the ground in part of a hyperbola. It hits every point on this curve at
the same time, so that people in different places along the curve on
the ground hear it at the same time. Because the airplane is moving
forward, the hyperbolic curve moves forward and eventually the
boom can be heard by everyone in its path. A hyperbola revolving
around its axis forms a surface called a hyperboloid. The cooling
tower of a steam power plant has the shape of a hyperboloid, as
does the architecture of the James S. McDonnell Planetarium of the
St. Louis Science Center.
Properties of hyperbolas
 Most calculus students have learned of the "reflecting
properties" of the parabola and the ellipse. If a "beam
of light" emanates from the focus of a parabola in any
direction, and is "reflected" from the parabola, it
subsequently travels in a line parallel to the axis of the
parabola. For the ellipse, a beam emanating from a
focus is reflected by the curve through the other focus.
 Less known is a reflecting property of hyperbolae. A
beam of light is directed at one of the foci (with the
the curve "between" the source and the focus) then it
will be reflected by the curve through the other focus!
 This property of the hyperbola is used in radio
direction finding (since the difference in signals from
two towers is constant along hyperbolae), and in the
construction of mirrors inside telescopes (to reflect
light coming from the parabolic mirror to the
eyepiece).
Chapter Objectives
 Now that we have explored conic sections in real life, lets look at
each one. At the end of this chapter, we will be able to do the
following:
 1. Use the standard and general forms of the equations of circles,
parabolas, ellipses, and hyperbolas.
 2. Graph circles, parabolas, ellipses and hyperbolas.
 3. Find the eccentricity of conic sections.
 4. Recognize conic sections by their equations.
 5. Find parametric equations for conic sections defined by
rectangular equations and vice versa.
 6. Graph and solve systems of second degree equations and
inequalities.
The Standard and General Forms of
Conics
 Standard form of conics:
 circle: (x-h) + (y-k) = r
 Ellipse:
bx  hg
by  kg
+
2
2
b2
a2
=1
 Parabola: (y-k) = 4p (x-h)
bx  hg - by  k g
2
 Hyperbola:
a2
(h,k) = center
b2
(h,k) = center
or
(x-h) = 4p ( y-k)
2
= 1 or
b g - bx  hg
yk
a2
2
b2
(h,K) = vertex
2
=1
(h,k) = center
 Rectangular Hyperbola: xy = k
 General Form of all conics: Ax + Bxy + Cy + Dx + Ey + F = 0





For a circle A=C and B=0
For an ellip se E ith er A or C is zero (W h y can ’t b oth b e zero?) an d B = 0
For a p arab ola A an d C h ave th e sam e sig n , A ≠ C , B = 0
For a hyperbola A and C have opposite signs B=0
For a rectangular hyperbola A=C=D=E=0
The Circle
 Any situation where a distance is known, but
not a direction can be modeled by a circle.
 Example # 1 : Write the standard form of the equation of the
circle that is tangent to the x axis and has its center at (-5,4).
Then graph the equation.
2
2
 Standard form (x+5) + (y-4) = 16
X
-5
4
Example #2
2
2
 The equation of a circle is 4x + 4y -24x +16y = -51
 1. Write the standard form of the equation
 2. Find the radius and the coordinates of the center.
 3. Graph the equation.
 To write the equation in standard form, we need to use our
knowledge of completing the square.
 Step 1: Divide everything through by 4 and we get
2
2
x + y -6x + 4y =  5 1  5 1
2
2
4
(x -6x) + (y +4y) =
4
(x-3) 2 + (y+2) 2
=  5 1 +9+4 =.025
4
Standard form: (x-3) 2 + (y+2) 2 = .25
Radius .5 center (3,-2)
Example #3
 From geometry, you know that any two points in the
coordinate plane determine a unique line. It is also true that
any three Noncollinear points in the coordinate plane
determine a unique circle. The equation of this circle can be
found by substituting the coordinates of the three points
into the general form of the equation of a circle and solving
the resulting system of three equations.
 Example: Write the standard form of the equation of the
circle that passes through the points at (1,1) (1,2) and
(2,3). Then identify the center and the radius of the circle.
 We have a system of three equations:
2
2
(1) + (1) +
D (1) + E (1) + F = 0
2
2
(1) + (2) + D (1) + E (2) + F = 0
2
2
(2) + (3) + D (2) + E (3) + F = 0
D+E+F+2=0
D+2E+F+5=0
2D+3E+F+13=0
 D+E+F+2=0
E=-3
 D+2E+F+5=0
 2D+3E+F+13=0
D+E=-8
D=-5
 Therefore F = 9
 So the general form of the equation is
2
2
x + y -5x -3y+9=0
Completing the 2square, we
have the standard
2
3
5
form as (x- 2 ) + (y- 2 ) = 52
Center (2.5, 1.5) radius
2.5
HW# 14
: Section 10-2
Pp. 627-630
#15-25 odds, 29,35,47