4.1%20Circles%20The%20First%20Conic%20Section - PowerPoint PPT Presentation

Title: 4.1%20Circles%20The%20First%20Conic%20Section


1
4.1 CirclesThe First Conic Section

• The basics, the definition, the formula
• And Tangent Lines

2
The Standard Form of a circle comes from the
Distance Formula

• What is the length of the hypotenuse below?

y2
y1
x1
x2
3
The Circle

• Definition The set of all points in a plane a
  given distance _________________
• from a given point ______________

4
The Circle

• The Formula

____________________________________________
5
Practice problems

• Find the center and radius of the following

Hint Complete the Square
6
How to graph?

• This is probably the easiest part of it all.
  Essentially, plot the center then
• r
• And then label those 4 points. If r is
  irrational, then estimate its value and plot in
  the general correct area (and label with radical
  notation)

7
Notes

• _____________________________________
• _____________________________________
• ________________________________________
• _____________________________________
• ________________________________________
• ________________________________________
• ________________________________________

8
Graph
9
Now we will learn to find the equation of their
tangent lines

• What do you know about a tangent line?
• _________________
• So, if it perpendicular to the radius, what do
  you know about the slopes.
• ___________________________

10
What does a tangent line look like?
11
So, what are the steps?

1. ________________________________
2. ________________________________
3. __________________________________________________
   ______________
4. __________________________________________________
   __________________________________________________
   __________________________________________________
   __________

12
Find the equation of the line tangent to the
circle
at (2,2)
13
Group Problems

1. Find the equation of the line tangent to the
   circle at
   (2,7).
2. Find the equation of the line tangent to the
   circle at the point in the
   4th quadrant where x 4.

14
10-1 Ellipses

• Fumbles and Kickoffs

15
The Definition

• (Dont write this down!!)
• The set of all points in a plane such that the
  sum of the distances from two fixed points,
  called foci (plural of focus) is a constant.

16
The Fumble
17

• The Kickoff

18
So, how are we going to tell which is which?
a
b
b
a
19
Thats right!!

• a gt b
• So, look for the larger number. If it is under
  x, _____________. If it is under the y,
  ______________________.
• When graphing, label center, major axis
  endpoints, minor axis endpoints and foci.

20
Foci, Where are they?

• They are on the major axis, c units from the
  center.
• How to find c?

21
Examples

• Graph the following completely.
• Remember Standard Form 1.

22
How to do this? Pull out any squared term
coefficients before completing the square. Then
divide so that the entire right side 1.
23
10-1 Ellipses

• Graph information to Equation
• (going the other direction)

24
Lets just look at this standard form, the fumble,
to see exactly what we need. You need the
______________________
25
Center

• You could be given
• ___________________________
• ___________________________
• ___________________________
• ___________________________

26
a

• You could be given
• _____________________________
• _____________________________
• _____________________________
• _____________________________

27
b

• You could be given
• ___________________________
• ___________________________
• ___________________________

28
Dont forget!

• You also need the orientation!! I would TRULY
  suggest always doing a quick little sketch to see
  the axes orientation.
• A longer vertical axis is a
• A longer horizontal axis is a

kickoff
fumble
29
Examples

• Center (1, 1) Focus (1, 3) Vertex (1, -9)

30
Fumble
2. Foci (4,2) and (8, 2) MA endpoints (3, 2),
(9, 2)
Use MP formula to find Center
Use
31
10-2 Hyperbolas

• Day 1
• Standard Equation and the Graph

32
The Definition

• The set of all points in a plane such that the
  difference of the distances from two points,
  called foci, is constant.
• Does that look familiar?

33
The Picture and Equation
34
The Other Orientation
Happy/Sad
35
Day 1 is simply drawing the figure

• You need to draw the center, the tranverse axis
  endpoints (still a) and the asymptotes. We will
  use the box method (more later on that) to
  make sure that the shape is accurate.
• Do you remember what the relationship between a,
  b and c was in ellipses?

36
How to find foci

• This time, you take the sum of the denominators

37
Lets go back to the definition
(h,k)

• The set of all points in a plane such that the
  difference of the distances from two points,
  called foci, is constant.

38
Remember

• With ellipses, you move the number under each
  variable in that direction. It will be a very
  similar method with hyperbolas.
• Now is when we introduce the
• Box Method

39
The Box Method

• Move a units away from the center in both
  directions to form the transverse axis endpoints.
• Move b units away from the TA EPs in both
  directions.
• _______________________________
• _______________________________
• _______________________________
• _______________________________

40
(No Transcript)
41
(No Transcript)
42
10-2 Hyperbolas

• Graph information to Equation
• (going the other direction)

43
Its the same process as before -How wonderful!!
You need the ____________________________
_ Does this change with a happy/sad? So, how
can we be given this information?
44
Center

• You could be given
• ___________________________
• ___________________________
• ___________________________

45
a

• You could be given
• ____________________________
• ____________________________
• ____________________________

46
b

• You could be given
• _____________________________________
• _____________________________________
• _____________________________________
• ________________________________________
• ________________________________________
  

47
Dont forget!

• You still need the orientation!! I would again
  suggest doing a quick little sketch to see the
  orientation.
• A vertical traverse axis is a
• A horizontal traverse axis is a

48
Examples

• Center (1, 3) TA ep (1, 7) Focus (1, -2)

49
2. TA eps (3, -3), (-5, -3) slope of asymptotes
50
10-3 Parabolas

• What? Again?

51
The definition

• The set of all points in a plane equidistant from
  a point (focus) and a line (directrix).

52
Parabola Up/Down
53
Left/Right
54
So what do we do with this?

1. _______________ and get it in conic form (but
   only if there is a linear term for x and/or y).
2. __________________________
3. Put _________, __________, and _______ on the
   graph. Label these only. No tables or plotting
   5 points this time. ?

55
Examples
1
-1
56
Examples
1
-1
57
6
-1
58
4. Find the equation of the parabola with focus
at (1, 5) and directrix at y 9
9
7
5
1