Locus Examples

ExamplesIn the figure, horizontal line L intersects the yaxis 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 \(y =  \frac{{{x^2}}}{6}  \frac{{2x}}{3} + \frac{5}{6}\).

ExamplesIn 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.
(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 \(y  1 = 0\).

ExamplesThe 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
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 \({x^2} + {y^2} = 121\).