A **locus** is the term used to describe the path of a single moving point that obeys certain conditions. The point A above will move around the origin at a distance 2 units away, depicted by the circle. Similarly, just like a compass, the path of a pencil is a circle with centre at the point of the compasses.

This study relates to distances between a point and a line, line and another line, or even a point to a point! Use the **distance** and **perpendicular distance** formula to solve problems relating to *loci* (plural form of *locus*).

### Examples

1. Find the equation of the locus of a point *P*(*x, y*) that moves so that is always 3 units from the origin.

### Show Answer

You may recognise this locus as a circle, centre (0, 0) radius 3 units. Its equation is given by .

Alternatively, use the **distance** formula.

Let *P*(*x, y*) be a point of the locus.

We want *PO = *3 (which is the distance *'d'* in the distance formula)

2. Find the equation of the locus of point *P*(*x, y*) that moves so that the distance *PA* to distance *PB* is in the ratio 2:1 where *A=*(-3, 1) and *B*=(2, -2).

### Show Answer

Let *P*(*x, y*) be a point of the locus.

Using Geogebra App, the trajectory of the locus is shown below:

3. Find the equation of the locus of point *P*(*x, y*) that is equidistant from a fixed point *A*(1, -2) and fixed line with equation *y=5*.

### Show Answer

## Circle as Locus

The circle with centre (*a, b*) and radius *r*, has the equation:

Hence, the circle with a centre at the origin has the equation:

### Examples:

1. Find the equation of the locus of a point that is always 2 units from the point (-1, 0).

### Show Answer

This is a circle with radius 2 and centre (-1, 0). Its equation is in the form:

2. Find the radius and the coordinates of the centre of the circle:

### Show Answer

This is a circle with centre (-1, 3) and radius 5.