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Mathematics Dictionary
Dr. K. G. Shih
Circle : Locus


  • Q01 | - Circle : Locus
  • Q02 | - Locus of x^2 + y^2 - 6*x + 6*y - 7 = 0
  • Q03 | - Locus of x^2 + y^2 - 6*x + 6*y + 18 = 0
  • Q04 | - Locus of x^2 + y^2 - 6*x + 6*y + 43 = 0
  • Q05 | - Locus of R = sin(A)
  • Q06 | - Locus of R = cos(A)
  • Q07 | - Locus of x = cos(t) and y = sin(t)

    • Q01. Circle : Locus of circle

    Question
    • C(h, k) is a fixed point
    • P(x, y) is a moving point
    • If PC = r is constant, what is the locus of point P ?
    Solution
    • Use distance formula
    • Sqr((x - h)^2 + (y - k)^2) = r
    • Square both sides
    • (x - h)^2 + (y - k)^2 = r^2
    • This is a circle with center at (h, k) and radius = r
    • This is the standard form of the equation of a circle
    Circle equation in implicity form
    • x^2 + y^2 + d*x + e*y + f = 0
    • Use completing the square we can change it to standard form

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    Q02. Locus of x^2 + y^2 - 6*x + 6*y - 7 = 0

    Solution
    • Completing the square
    • (x^2 - 6*x + 9 - 9) + (y^2 + 6*y + 9 - 9) - 7 = 0
    • (x - 3)^2 + (y + 3)^2 = 5^2
    • It is a circle with center at (3, -3) and radius = 5

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    Q03. Locus of x^2 + y^2 - 6*x + 6*y + 18 = 0

    Solution
    • Completing the square
    • (x^2 - 6*x + 9 - 9) + (y^2 + 6*y + 9 - 9) + 18 = 0
    • (x - 3)^2 + (y + 3)^2 = 0
    • It is a point (3, -3)
    • Hence there is no locus

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    Q04. Locus of x^2 + y^2 - 6*x + 6*y + 43 = 0

    Solution
    • Completing the square
    • (x^2 - 6*x + 9 - 9) + (y^2 + 6*y + 9 - 9) + 43 = 0
    • (x - 3)^2 + (y + 3)^2 = - 25
    • This is not existed in the real number system
    • Hence there is no locus

    Go to Begin

    Q05. Locus of R = sin(A)

    Question
    • What is the locus of R = sin(A)
    Solution
    • Polar coordinates
      • R = x^2 + y^2
      • x = R*cos(A)
      • y = R*sin(A)
    • Change polar form to rectangular
      • R = y/R
      • R^2 = y
      • x^2 + y^2 = y
      • Completing the square
      • x^2 + (y^2 - y + 1/4 - 1/4) = 0
      • x^2 + (y - 1/2)^2 = (1/2)^2
      • This is a circle with center at (0, 1/2) and radius = 1/2

    Go to Begin

    Q06. Locus of R = cos(A)

    Question
    • What is the locus of R = cos(A)
    Solution
    • Polar coordinates
      • R = x^2 + y^2
      • x = R*cos(A)
      • y = R*sin(A)
    • Change polar form to rectangular
      • R = x/R
      • R^2 = x
      • x^2 + y^2 = x
      • Completing the square
      • (x^2 - x + 1/4 - 1/4) + y^2 = 0
      • (x - 1/2)^2 + y^2 = (1/2)^2
      • This is a circle with center at (1/2, 0) and radius = 1/2

    Go to Begin

    Q07. Locus of x = cos(t) and y = sin(t)

    Question
    • Parametric equation : x = cos(t) and y = sin(t)
    • What is the locus of this equation
    Solution
    • x^2 + y^2 = cos(t)^2 + sin(t)^2
    • Since cos(t)^2 + sin(t)^2 = 1
    • Hence x^2 + y^2 = 1
    • This is circle with center at (0, 0) and radius = 1

    Go to Begin
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