### Locus 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 $${O_1}$$.
• (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.

• 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.

• 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 $$x \ne 3$$.

• 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.
• (a) 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 $$x + 2y - 2 = 0$$.