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9.3 – Circles with Coordinates and Proofs

Key Terms

  • Circle – All points on a plane that are the same distance from a single point.
  • Center of the Circle – The point exactly in the middle of a circle.
  • Origin – Where x and y coordinates are both zero on a coordinate plane.
    • Written as (0,0)
  • Radius – The distance from the center of a circle to any point on the circle.
    • The size of a circle is defined by its radius.

Review

 Solutions for Lines and Circles
  • The solution set for the equation of a line is all the points that lie on the line.
  • The solution set for the equation of a circle is all the points that lie on the circle.
  • Circle Formula and Coordinates

GeoB 9.2 - Circle Diagram

  • Unit Circle

GeoB 9.2 - Unit Circle

Notes

 Distance Formula: Used to Find the Radius of a Circle
  • D=\sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}
  • r=\sqrt{(x-0)^{2}+(y-0)^{2}}
  • r=\sqrt{x^{2}+y^{2}}
Standard Equation for Circle Centered at Origin Standard Equation for Any Circle
  • x^{2}+y^{2}=r^{2}
    • r = radius of the circle
    • x = x-coordinate of point on the circle
    • y = y-coordinate of point on the circle

 

  • Ex. Circle centered at the origin with radius 8
    • x^{2}+y^{2}=8^{2}
    • x^{2}+y^{2}=64
  • (x-h)^{2}+(y-v)^{2}=r^{2}
    • r = radius of the circle
    • (x, y) = a point on the circle
    • (h, v) = center of the circle
      • h: horizontal (x-coordinate)
      • v: vertical (y-coordinate)

 

  • Ex. Circle centered at (-3,5) with radius 5
    • (x-(-3))^{2}+(y-(5))^{2}=5^{2}
    • (x+3)^{2}+(y-5)^{2}=25
Moving the Circle Graphing and Writing the Equation of a Circle
  • When you move the circle left or right, the number that changes in its equation is the x-coordinate of the center.
  • When you move the circle up or down, the number that changes in its equation is the y-coordinate of the center.
  • To write the equation for any circle, you need to know its radius and the coordinates of its center.
  • You can use the equation of a circle to graph the circle on an xy-plane and find the coordinates of points on the circle.
Quadrants Circles on the Axes
  • Each xy-plane has 4 quadrants
    • Quadrant I
      • Top right: (+,+)
      • (x-h)^{2}+(y-v)^{2}=r^{2}
    • Quadrant II
      • Top left: (-,+)
      • (x+h)^{2}+(y-v)^{2}=r^{2}
    • Quadrant III
      • Bottom left: (-,-)
      • (x+h)^{2}+(y+v)^{2}=r^{2}
    • Quadrant IV
      • Bottom right: (+,-)
      • (x-h)^{2}+(y+v)^{2}=r^{2}
  • Circles Centered on the x-axis
    • (x-0)^{2}+(y-v)^{2}=r^{2}
    • (x-0)^{2}+(y+v)^{2}=r^{2}
  • Circles Centered on the y-axis
    • (x-h)^{2}+(y-0)^{2}=r^{2}
    • (x+h)^{2}+(y-0)^{2}=r^{2}

 

  • Ex. The following circles have their centers in the second quadrant.
    • (x+4)^{2}+(y-4)^{2}=7
    • (x+11)^{2}+(y-8)^{2}=18
  • Ex. The following circles have their centers in the third quadrant.
    • (x+16)^{2}+(y+3)^{2}=17
    • (x+9)^{2}+(y+12)^{2}=36

Examples

  • Ex 1. This circle is centered at the origin, and the length of its radius is 6. What is the circle’s equation?
    • Answer: x^{2}+y^{2}=36

GeoB 9.2 - Q1-1c

  • Ex 2. The equation for the circle below is x^{2}+y^{2}=81. What is the length of the circle’s radius?
    • Answer: 9

GeoB 9.2 - Q2-2c

  • Ex 3. The circle below is centered at the point (4, 1) and has a radius of length 2. What is its equation?
    • Answer: (x-4)^{2}+(y-1)^{2}=4

GeoB 9.2 - Q2-1c

  • Ex 4. The circle below is centered at the point (4, -3) and has a radius of length 5. What is its equation?
    • Answer: (x-4)^{2}+(y+3)^{2}=5^{2}
  • Ex 5. What is the equation of the circle with center (-3.2, -2.1) and radius 4.3?
    • Answer: (x+3.2)^{2}+(y+2.1)^{2}=4.3^{2}
  • Ex 6. The following circles lie completely within the third quadrant, not just the center.
    • Graph them to find out for sure.
      • (x+12)^{2}+(y+9)^{2}=9
      • (x+7)^{2}+(y+7)^{2}=4

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