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Understanding Distance and Midpoint Formulas in Analytic Geometry

This chapter delves into the fundamental concepts of distance and midpoint formulas, relating them to the Pythagorean theorem. It explores how to calculate the distance between two points in a coordinate plane and find the midpoint of a segment. Furthermore, it covers conics, including circles, parabolas, ellipses, and hyperbolas, discussing their equations and properties. The text includes examples illustrating the application of these formulas and examples for deriving equations for circles, parabolas, and ellipses based on given parameters.

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Understanding Distance and Midpoint Formulas in Analytic Geometry

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  1. Chapter 9 Analytic Geometry

  2. Section 9-1 Distance and Midpoint Formulas

  3. Pythagorean Theorem • If the length of the hypotenuse of a right triangle is c, and the lengths of the other two sides are a and b, then c2 = a2 + b2

  4. Example Find the distance between point D and point F.

  5. Distance Formula D = √(x2 – x1)2 + (y2 – y1)2

  6. Example • Find the distance between points A(4, -2) and B(7, 2) • d = 5

  7. Midpoint Formula M( x1 + x2, y1 + y2) 2 2

  8. Example • Find the midpoint of the segment joining the points (4, -6) and (-3, 2) • M(1/2, -2)

  9. Section 9-2 Circles

  10. Conics • Are obtained by slicing a double cone • Circles, Ellipses, Parabolas, and Hyperbolas

  11. Equation of a Circle The circle with center (h,k) and radius r has the equation (x – h)2 + (y – k)2 = r2

  12. Example • Find an equation of the circle with center (-2,5) and radius 3. • (x + 2)2 + (y – 5)2 = 9

  13. Translation • Sliding a graph to a new position in the coordinate plane without changing its shape

  14. Translation

  15. Example Graph (x – 2)2 + (y + 6)2 = 4

  16. Example • If the graph of the equation is a circle, find its center and radius. • x2 + y2 + 10x – 4y + 21 = 0

  17. Section 9-3 Parabolas

  18. Parabola • A set of all points equidistant from a fixed line called the directrix, and a fixed point not on the line, called the focus

  19. Vertex • The midpoint between the focus and the directrix.

  20. Parabola - Equations y-k = a(x-h)2 Vertex (h,k) symmetry x = h x - h = a(y-k)2 Vertex (h,k) symmetry y = k

  21. Equation of a Parabola • Remember: y – k = a(x – h)2 • (h,k) is the vertex of the parabola

  22. Example 1 • The vertex of a parabola is (-5, 1) and the directrix is the line y = -2. Find the focus of the parabola. • (-5 4)

  23. Example 1

  24. Example 2 • Find an equation of the parabola having the point F(0, -2) as the focus and the line x = 3 as the directrix.

  25. y – k = a(x – h)2 • a = 1/4c where c is the distance between the vertex and focus • Parabola opens upward if a>0, and downward if a< 0

  26. y – k = a(x – h)2 • Vertex (h, k) • Focus (h, k+c) • Directrix y = k – c • Axis of Symmetry x = h

  27. x - h = a(y –k)2 • a = 1/4c where c is the distance between the vertex and focus • Parabola opens to the right if a>0, and to the left if a< 0

  28. x – h = a(y – k)2 • Vertex (h, k) • Focus (h + c, k) • Directrix x = h - c • Axis of Symmetry y = k

  29. Example 3 Find the vertex, focus, directrix , and axis of symmetry of the parabola: y2 – 12x -2y + 25 = 0

  30. Example 4 Find an equation of the parabola that has vertex (4,2) and directrix y = 5

  31. Section 9-4 Ellipses

  32. Ellipse • The set of all points P in the plane such that the sum of the distances from P to two fixed points is a given constant.

  33. Focus (foci) • Each fixed point • Labeled as F1 and F2 • PF1 and PF2 are the focal radii of P

  34. Ellipse- major x-axis

  35. Ellipse- major y-axis

  36. Example 1 • Find the equation of an ellipse having foci (-4, 0) and (4, 0) and sum of focal radii 10. Use the distance formula.

  37. Example 1 - continued • Set up the equation PF1 + PF2 = 10 √(x + 4)2 + y2 + √(x – 4)2 + y2 = 10 • Simplify to get x2 + y2 = 1 25 9

  38. Graphing • The graph has 4 intercepts • (5, 0), (-5, 0), (0, 3) and (0, -3)

  39. Symmetry • The ellipse is symmetric about the x-axis if the denominator of x2 is larger and is symmetric about the y-axis if the denominator of y2 is larger

  40. Center • The midpoint of the line segment joining its foci

  41. General Form x2 + y2 = 1 a2 b2 The center is (0,0) and the foci are (-c, 0) and (c, 0) where b2 = a2 – c2 focal radii = 2a

  42. General Form x2 + y2 = 1 b2 a2 The center is (0,0) and the foci are (0, -c) and (0, c) where b2 = a2 – c2 focal radii = 2a

  43. Finding the Foci • If you have the equation, you can find the foci by solving the equation b2 =a2 – c2

  44. Example 2 Graph the ellipse 4x2 + y2 = 64 and find its foci

  45. Example 3 • Find an equation of an ellipse having x-intercepts √2 and - √2 and y-intercepts 3 and -3.

  46. Example 4 • Find an equation of an ellipse having foci (-3,0) and (3,0) and sum of focal radii equal to 12.

  47. Section 9-5 Hyperbolas

  48. Hyperbola • The set of all points P in the plane such that the difference between the distances from P to two fixed points is a given constant.

  49. Focal (foci) • Each fixed point • Labeled as F1 and F2 • PF1 and PF2 are the focal radii of P

  50. Example 1 • Find the equation of the hyperbola having foci (-5, 0) and (5, 0) and difference of focal radii 6. Use the distance formula.

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