Locus Examples


  • Graph
    [GraphMissing Latex_Locus Q38]
  • Examples
    In the figure, horizontal line L intersects the y-axis at (0, 3). If a moving point P is equidistant from A(-2, 0) and horizontal line L, find the equation of the locus of P.
  • Solutions

    Let (x, y) be the coordinates of the moving point P.
    \(\begin{array}{1}AP = \sqrt {{{[x - ( - 2)]}^2} + {{(y - 0)}^2}} {\rm{ u n i t s}}\\ = \sqrt {{{(x + 2)}^2} + {y^2}} {\rm{ u n i t s}}\end{array}\)
    ∵ P lies below horizontal line L.
    ∴ Distance PN of P from horizontal line L\( = (3 - y){\rm{ u n i t s}}\)
    ∵ \(AP = PN\)
    ∴ \(\begin{array}{1}\sqrt {{{(x + 2)}^2} + {y^2}} = 3 - y\\{(x + 2)^2} + {y^2} = {(3 - y)^2}\\{x^2} + 4x + 4 + {y^2} = 9 - 6y + {y^2}\\{x^2} + 4x - 5 = - 6y\\y = - \frac{{{x^2}}}{6} - \frac{{2x}}{3} + \frac{5}{6}\end{array}\)
    ∴ The equation of the locus of P is \(y = - \frac{{{x^2}}}{6} - \frac{{2x}}{3} + \frac{5}{6}\).


  • Graph
    [GraphMissing Latex_Locus Q54]
  • Examples
    In the figure, the coordinates of M are (1, 3) and the coordinates of N are (5, 3). P is a moving point such that the area of ΔMPN is 4 square units.
    (a) Draw the locus of P.
    (b) Find the equation(s) of the locus of P.
  • Solutions

    (a) Let h units be the height of ΔMPN with the base of MN.
    \(MN = (5 - 1)\) units
    \( = 4\) units
    ∵ Area of ΔMPN = 4 square units
    ∴ \(\begin{array}{c}\frac{1}{2} \times 4 \times h = 4\\h = 2\end{array}\)
    ∴ The locus of P is shown in the figure.
    [GraphMissing Latex_Locus Q54]

    (b) Let (x, y) be the coordinates of the moving point P.
    When P lies above MN,
    distance of P from MN\( = (y - 3)\) units
    \(\begin{array}{ccccc}\therefore {\rm{ }}y - 3 = & \;2\\y - 5 = & \;{0^{}}\end{array}\)
    When P lies below MN,
    distance of P from MN\( = (3 - y)\) units
    \(\begin{array}{ccccc}\therefore {\rm{ }}3 - y = & \;2\\y - 1 = & \;{0^{}}\end{array}\)
    ∴ The equations of the locus of P are \(y - 5 = 0\) and \(y - 1 = 0\).


  • Graph
    [GraphMissing Latex_Locus Q56]
  • Examples
    The figure shows a circle with the centre at the origin O and the radius of 9 units. A moving point P keeps a fixed distance of 2 units from the circumference of the circle. Find the equation(s) of the locus of P.
  • Solutions

    [GraphMissing Latex_Locus Q56]
    As shown in the figure, the locus of P is two circles with the same centre at O and the radii of 7 units and 11 units individually.
    i.e. P keeps fixed distances of 7 units and 11 units from O.
    Let (x, y) be the coordinates of the moving point P.
    When P keeps a fixed distance of 7 units from O,
    \(\begin{array}{1}\sqrt {{{(x - 0)}^2} + {{(y - 0)}^2}} = 7\\{x^2} + {y^2} = 49\end{array}\)
    When P keeps a fixed distance of 11 units from O,
    \(\begin{array}{1}\sqrt {{{(x - 0)}^2} + {{(y - 0)}^2}} = 11\\{x^2} + {y^2} = 121\end{array}\)
    ∴ The equations of the locus of P are \({x^2} + {y^2} = 49\) and \({x^2} + {y^2} = 121\).