 7.6 Equation of a Locus

7.6 Equation of a Locus
1. The equation of the locus of a moving point P(x, y) which is always at a constant distance (r) from a fixed point (x1, y1) is:

2. The equation of the locus of a moving point P(x, y) which is always at a constant distance from two fixed points (x1, y1) and  (x1, y1) with a ratio is:

3. The equation of the locus of a moving point P (x, y) which is always equidistant from two fixed points A and B is the perpendicular bisector of the straight line AB.

Example 1
Find the equation of the locus of a moving point P(x, y) which is always at a distance of 5 units from a fixed point Q (2, 4).

Solution:
(xx1)2+ (yy1)2 = r2
(x – 2)2 + (y – 4)2 = 52
x2 – 4x + 4 + y2 – 8y + 16 = 25
x2 + y2– 4x – 8y – 5 = 0

Example 2
Find the equation of the locus of a moving point P(x, y) which is always equidistant from points A (-2, 3) and B (4, -1).

Solution:

Example 3
A (2, 0) and B (0, -2) are two fixed points. Point P moves with a ratio so that APPB = 1: 2.  Find the equation of the locus of point P.

Solution: