Locus Examples

ExamplesIn 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.

ExamplesIt 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 \(x \ne 3\).

ExamplesIn 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) 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 \(x + 2y  2 = 0\).