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Conic Sections: The Circle

Discover the essential concepts of circles, from geometric definitions to algebraic equations. Understand how to find the center and radius, and differentiate circles from lines and parabolas. Practice writing equations and sketching circles in center-radius form.

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Conic Sections: The Circle

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  1. Conic Sections:The Circle Colleen Beaudoin February, 2009

  2. Circle • 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).

  3. Circle 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. (x,y) r (h,k) x y

  4. Items referenced on the graph of a circle: Radius (r) Center (h,k)

  5. Facts: Circle Equation • 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?

  6. Find the center and radius for each of the following circles.

  7. 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

  8. 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:

  9. 8. Write in center-radius form and sketch: Hint: You must complete the square.

  10. Review • 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?

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