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Unit 5 Conic Sections

Unit 5 Conic Sections. The Circle. History. Conic sections is one of the oldest math subjects studied. The conics were discovered by a Greek mathematician named Menaechmus (c. 375-325 BC). Menaechmus’s intelligence was highly regarded… he tutored Alexander the Great. Appollonius.

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Unit 5 Conic Sections

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  1. Unit 5 Conic Sections The Circle

  2. History • Conic sections is one of the oldest math subjects studied. • The conics were discovered by a Greek mathematician named Menaechmus (c. 375-325 BC). • Menaechmus’s intelligence was highly regarded… he tutored Alexander the Great.

  3. Appollonius History • Appollonius (c. 262-190 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.

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

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

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

  7. Example 2 Center: Radius:

  8. Example 3 Center ? Radius ?

  9. 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?

  10. Find the center and radius of the circle:

  11. Example

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

  13. Warm up • Find the center and radius of the following circles

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

  15. MINI QUIZ • Find the center and radius of the following circles

  16. TIME TO GRAPH!

  17. 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

  18. Your turn...

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

  20. Circles Continued

  21. 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

  22. Example

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

  24. 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

  25. 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 = -1.56. Using the Quadratic Formula x = Substituting these values into the linear equation yields y = -1.56 and y = 2.56 respectively. Therefore the line intersects the circle in two points (2.56, -1.56) and (-1.56, 2.56).

  26. 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.

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

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

  29. Example

  30. H W • Worksheet: Conic Sections: Circle

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