Locus Examples


  • Examples
    In the figure, \({O_1}\) and \({O_2}\) are the centres of a small circle and a large circle respectively, and \({O_1}{O_2} = 2{\rm{ cm}}\). The small circle touches the large circle internally. Sketch and describe the locus of a moving point which satisfies each of the following conditions.
    (a) \({O_2}\) is a fixed point and the moving point \({O_1}\) keeps a fixed distance from \({O_2}\).
    (b) \({O_1}\) is a fixed point and the moving point \({O_2}\) keeps a fixed distance from \({O_1}\).
  • Solutions

    (a) The locus of \({O_1}\) is a circle with the centre at \({O_2}\) and the radius of 2 cm.



    (b) The locus of \({O_2}\) is a circle with the centre at \({O_1}\) and the radius of 2 cm.



  • Graph
  • Examples
    It is given that A(-4, 0) and B(3, -7) are two fixed points. If P is a moving point such that PA ⊥ PB, find the equation of the locus of P.
  • Solutions

    Let (x, y) be the coordinates of the moving point P.
    Slope of PA\( = \frac{{y - 0}}{{x - ( - 4)}} = \frac{y}{{x + 4}}\)
    Slope of PB\( = \frac{{y - ( - 7)}}{{x - 3}} = \frac{{y + 7}}{{x - 3}}\)
    ∵ PA ⊥ PB
    ∴ \(\begin{array}{1}\frac{y}{{x + 4}} \cdot \frac{{y + 7}}{{x - 3}} = - 1\\y(y + 7) = - (x + 4)(x - 3)\\{y^2} + 7y = - ({x^2} + x - 12)\\{y^2} + 7y = - {x^2} - x + 12\\{x^2} + {y^2} + x + 7y - 12 = 0\end{array}\)
    ∴ The equation of the locus of P is \({x^2} + {y^2} + x + 7y - 12 = 0\), where \(x \ne - 4\) and \(x \ne 3\).


  • Graph
    [GraphMissing Latex_Locus Q50]
  • Examples
    In the figure, the equation of straight line L is \(x + 2y - 6 = 0\). A moving point P lies on straight line L, and Q(x, y) is a moving point such that \(OQ:QP = 1:2\).
    (a) Draw and describe the locus of Q.
    (b) (i) Express the coordinates of P in terms of x and y.
         (ii) Find the equation of the locus of Q.
  • Solutions

    (a)[GraphMissing Latex_Locus Q50]
    The locus of Q is a straight line below L.

    (b) (i) Let \(({x_1},{\rm{ }}{y_1})\) be the coordinates of P.
    \(\begin{array}{1}x = \frac{{{x_1} + 2 \times 0}}{{1 + 2}}\\{x_1} = 3x\\y = {\frac{{{y_1} + 2 \times 0}}{{1 + 2}}^{}}\\{y_1} = 3y\end{array}\)
    ∴ The coordinates of P are (3x, 3y).
    (ii) ∵ P lies on straight line L.
    ∴ \(\begin{array}{1}{x_1} + 2{y_1} - 6 = 0\\3x + 2(3y) - 6 = 0\\x + 2y - 2 = 0\end{array}\)
    ∴ The equation of the locus of Q is \(x + 2y - 2 = 0\).