1. Warm Up
Complete the square 1) 2) 3)

2. Unit 5 Conic Sections
The Circle

3. History Conic sections is one of the oldest math subjects studied.
The conics were discovered by a Greek mathematician named Menaechmus (c BC).
Menaechmus’s intelligence was highly regarded… he tutored Alexander the Great.

4. Appollonius
History
Appollonius (c BC) wrote about conics in his series of books simply titled “Conic Sections”.
Appollonious’ nickname was “the Great Geometer”
He was the first to base the theory of all three conics on sections of one circular cone.
He is also the one to assign the name “ellipse”, “parabola”, and “hyperbola” to three of the conic sections.

5. A conic section is a curve formed by the intersection of _________________________
a plane and a double cone.

6. Circles
The set of all points that are the same distance from the center.
Standard Equation:
(h , k) r
With CENTER: (h, k)
& RADIUS: r (square root)

7. Example 1 -h r² -k
Center:
Radius: r ( ) , h k

8. Example 2
Center:
Radius:

9. Example 3
Center ?
Radius ?

10. Not In Standard form?
Move all variables to one side (group like terms together) and the constant to the other side
Complete the square on both variables to put it in standard form
Factor as squares EX
Center? Radius?

11. Find the center and radius of the circle:

12. Example

13. HW: Write the Equation of the circle in S.F by completing the square

14. Warm up
Find the center and radius of the following circles

15. Warm up
Find the center and radius of the following circles

16. MINI QUIZ
Find the center and radius of the following circles

17. TIME TO GRAPH!

18. Center (x, y):
(0, 0)
Radius(r):
To graph:
1.) plot the center coordinate
2.) go up, down, left & right r units
3.) Sketch a circle using these points as guides

19. Your turn...

20. Write the equation for the given information and then graph.
4.) Center at (5, -2) and a radius of 4

21. Circles Continued

22. Tangent line and a Circle
In order to write the equation of a line, we need a point on the line and the slope. In this case, we know the slope of the radius is Since the tangent line is perpendicular to the radius, the slope of the tangent line must be
1st Rewrite the equation as
2nd Insert your given point for x and y
3rd Solve the new equation for y and put in y = mx + b form

23. Example

24. Systems of Equations Containing a Circle and a Line
We solve them using graphing and substitution.
Three possible solutions
No solution-they don’t intersect at all
One solution-they intersect at exactly one point
Two solutions-they intersect at two points

25. Solving Graphically Graph the circle using the center and the radius
Solve the linear equation for “y” and graph the line using the slope and y-intercept
Center: (0,0) r = 7
m = 1, y-int = -7

26. Given a system of equations such as
Solve y in terms of x and substitute into the circle equation.
y = -x + 1
x2 + (-x + 1)2 = 9
x2 + x2  2x + 1 = 9
2x2  2x  8 = 0
x2  x  4 = 0
, which gives x = 2.56 and x =
Using the Quadratic Formula x =
Substituting these values into the linear equation yields y = and y = 2.56 respectively. Therefore the line intersects the circle in two points (2.56, -1.56) and
(-1.56, 2.56).

27. Warm up: Choose one and solve

28. To check solutions by graphing on the TI-83 Calculators
Solve both equations for y
Press “y=“ and enter the equations in y1, y2, and y3
Press window. Your graphing window should be a x to y ratio of 3 to 2 (ex. 9 to 6 or 12 to 8)
Press graph
To find the solutions of where they intersect, Press “Zoom” “1”. Use the arrows to get close to an intersection. Press “enter”. Use the arrows to open your box to surround your intersection. Press “enter”. Press “trace” and use the arrows to move to the intersection point. Your solution will be at the bottom of the screen as x and y.

29. System of Equations with Two Circles
No Solution
One solution
Two Solutions
We solve these by graphing

30. Graphically
Graph both circles on the same set of axes and find the intersection points

31. Algebraically
Multiply one equation by -1 and add the two equations (this will eliminate the x2, and y2
Solve the remaining equation for x or y
Substitute this equation in an original equation and solve
Substitute the answer into the equation from Step two to find the other coordinate

32. To find solutions by graphing on the TI-83 Calculators
Solve both equations for y (you will have 4 equations)
Press “y=“ and enter the equations in y1, y2, y3 and y4
Press window. Your graphing window should be a x to y ratio of 3 to 2 (ex. 9 to 6 or 12 to 8)
Press graph
To find the solutions of where they intersect, Press “Zoom” “1”. Use the arrows to get close to an intersection. Press “enter”. Use the arrows to open your box to surround your intersection. Press “enter”. Press “trace” and use the arrows to move to the intersection point. Your solution will be at the bottom of the screen as x and y.

33. Example

34. H W
Worksheet: Conic Sections: Circle