### Locus 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.

• 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 $$y = - \frac{{{x^2}}}{6} - \frac{{2x}}{3} + \frac{5}{6}$$.

• 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.
• (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 $$y - 1 = 0$$.

• 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.

• 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 $${x^2} + {y^2} = 121$$.