Circles
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  • Discuss how the radius can be drawn from the center to any point on the circle.
  • Stress that in a linear equation, no variables are squared. In the parabola equation only one variable is squared, while two are squared in the circle equation.
  • Ax+BY=C Stress that linear equations have exponents of 1 on both the x and the y. With parabolas, either the x or the y has an exponent of 2. With circles, both x and y terms are squared. Also with circles, the coefficients (or denominators) are the same for x 2 and y 2 (You will see in another lesson that that is not true for ellipses.)

Circles Presentation Transcript

  • 1. Next CONIC SECTIONS Parabola Circle Ellipse Hyperbola Quadratic Relations Previous Main Menu End
  • 2. What are conics?
    • Conics, or conic sections, are the intersection of a plane with an infinite double cone. If that plane cuts both cones, it is a hyperbola. If it is parallel to the edge of the cone, you get a parabola. If neither is the case, it is an ellipse. The ellipse is also a circle if the plane is perpendicular to the altitude of the cone.
  • 3. Circle ©National Science Foundation
  • 4. Circle
    • The Standard Form of a circle with a center at (0,0) and a radius, r, is……..
                                                                           center (0,0) radius = 2 Copyright ©1999-2004 Oswego City School District Regents Exam Prep Center
  • 5. Circles
    • The Standard Form of a circle with a center at (h,k) and a radius, r, is……..
                                                                                                                                                                                center (3,3) radius = 2 Copyright ©1999-2004 Oswego City School District Regents Exam Prep Center
  • 6.
    • Review: The geometric definition relies on a cone and a plane intersecting it
    • Algebraic definition: a set of points in the plane that are equidistant from a fixed point on the plane (the center).
  • 7.
    • Find the distance from the center of the circle (h,k) to any point on the circle (represented by (x,y)). This is the radius of the circle.
    • Review the distance formula:
    • Substitute in the values.
    • Square both sides to get
    • the general form of a
    • circle in center-radius form.
    y x r (h,k) (x,y)
  • 8.
    • Radius (r)
    • Center (h,k)
  • 9.
    • Both variables are squared.
    • Equation of a circle in center-radius form:
    • What makes the circle different from the a line?
    • What makes the circle different from the parabola?
  • 10.  
  • 11.
    • 4. Write the equation of a circle centered at (2,-7) and having a radius of 5.
    • (x - 2) 2 + (y + 7) 2 = 25
    • 5. Describe (x - 2) 2 + (y + 1) 2 = 0
    • A point at (2,-1)
    • 6. Describe (x + 1) 2 + (y - 3) 2 = -1
    • No graph
  • 12.
    • 7. Write the equation of a circle whose diameter is the line segment joining A(-3,-4) and B(4,3).
    • What must you find first?
    • The center and the radius.
    • How can you find the center?
    • The center is the midpoint of the segment.
    • (½ , - ½ )
    • How can you find the radius?
    • The radius is the distance from the center to a point on the circle. Use the distance formula.
    • The equation is:
  • 13.
    • 8. Write in center-radius form and sketch:
    • Hint: You must complete the square.
  • 14.
    • What’s the standard form of a line?
    • What are the steps for graphing a circle?
    • How can you tell if the graph of an equation will be a line, a parabola, or a circle?