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Analytic Geometry: Lines and Circles · Ages 16–17

Finding the Center and Radius of a Circle

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For 11th grade geometry students. Determine the center and radius for each circle equation, then classify the given points.

  1. Identify the center and radius: (x - 4)^2 + (y - 7)^2 = 49.

  2. Identify the center and radius: (x + 2)^2 + (y - 9)^2 = 20.

  3. Identify the center and radius: x^2 + (y + 3)^2 = 64.

  4. Convert to standard form and state the center and radius: x^2 + y^2 - 8x - 10y + 5 = 0.

  5. Convert to standard form and state the center and radius: x^2 + y^2 + 12x - 4y - 9 = 0.

  6. A circle has equation (x - 1)^2 + (y + 2)^2 = 25. Determine whether the point (4, 2) lies inside, on, or outside the circle.

  7. A diameter of a circle connects endpoints A(-3, 2) and B(5, 8). Find the center and radius of this circle.

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Finding the Center and Radius of a Circle

Answer key

  1. Identify the center and radius: (x - 4)^2 + (y - 7)^2 = 49.

    Center: (4, 7), Radius: 7

  2. Identify the center and radius: (x + 2)^2 + (y - 9)^2 = 20.

    Center: (-2, 9), Radius: sqrt(20) or 2*sqrt(5)

  3. Identify the center and radius: x^2 + (y + 3)^2 = 64.

    Center: (0, -3), Radius: 8

  4. Convert to standard form and state the center and radius: x^2 + y^2 - 8x - 10y + 5 = 0.

    (x - 4)^2 + (y - 5)^2 = 36. Center: (4, 5), Radius: 6

  5. Convert to standard form and state the center and radius: x^2 + y^2 + 12x - 4y - 9 = 0.

    (x + 6)^2 + (y - 2)^2 = 49. Center: (-6, 2), Radius: 7

  6. A circle has equation (x - 1)^2 + (y + 2)^2 = 25. Determine whether the point (4, 2) lies inside, on, or outside the circle.

    On the circle. (4 - 1)^2 + (2 + 2)^2 = 3^2 + 4^2 = 9 + 16 = 25, which matches r^2.

  7. A diameter of a circle connects endpoints A(-3, 2) and B(5, 8). Find the center and radius of this circle.

    Center is the midpoint (1, 5). Radius is the distance from center to B: sqrt((5-1)^2 + (8-5)^2) = sqrt(16 + 9) = 5.