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Conic Sections

Chapter 10. Conic Sections. Chapter Sections. 10.1 – The Parabola and the Circle 10.2 – The Ellipse 10.3 – The Hyperbola 10.4 – Nonlinear Systems of Equations and Their Applications. The Parabola and the Circle. § 10.1. Identify and Describe the Conic Sections.

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Conic Sections

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  1. Chapter 10 Conic Sections

  2. Chapter Sections 10.1 – The Parabola and the Circle 10.2 – The Ellipse 10.3 – The Hyperbola 10.4 – Nonlinear Systems of Equations and Their Applications

  3. The Parabola and the Circle § 10.1

  4. Identify and Describe the Conic Sections Types of conic sections include parabolas, circles, ellipses, and hyperbolas.

  5. Graph Parabolas of the Form x = a(y – k)2 + h Parabola with Vertex at (h, k) y = a(x – h)2 + k, a > 0 (opens upward) y = a(x – h)2 + k, a < 0 (opens downward) y = a(y – k)2 + h, a > 0 (opens to the right) y = a(y – k)2 + h, a < 0 (opens to the left)

  6. Graph Parabolas of the Form x = a(y – k)2 + h

  7. Graph Parabolas of the Form x = a(y – k)2 + h Example Sketch the graph of x = -2(y + 4)2 – 1. The graph opens to the lefts since the equation is of the form x = a(y-k)2 + h and a = -2. h = -1 and k = -4, so the vertex is (-1, -4).

  8. Learn the Distance and Midpoint Formulas The horizontal distance between the two points (x1, y1) and (x2, y2), indicated by the blue dashed line is |x2 – x1| and the vertical distance indicated by the red dashed line is |y2 – y1|.

  9. Learn the Distance and Midpoint Formulas Distance Formula The distance, d, between any two points (x1, y1) and (x2, y2) can be found by the distance formula:

  10. Learn the Distance and Midpoint Formulas Midpoint Formula Given any two points (x1, y1) and (x2, y2), the point halfway between the given points can be found by the midpoint formula:

  11. Graph Circles with Centers at the Origin Circle A circle is the set of points in a plane that are the same distance, called the radius, from a fixed point, called the center. Circle with Its Center at the Origin and Radius r

  12. Graph Circles with Centers at the Origin For example, x2 + y2 = 16 is a circle with its center at the origin and radius 4, and x2 + y2 = 10 is a circle with its center at the origin and radius √10.

  13. Graph Circles with Centers at (h, k) Circle with Its Center at (h, k) and Radius r

  14. Graph Circles with Centers at (h, k) Example Determine the equation of the circle shown below. The center is (-3, 2) and the radius is 3.

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