Locus

Ken Grant, Loyola College

A line may be regarded as the path in which a point moves, or more strictly as the 'locus' of a point, locus meaning the path which is formed by the motion of anything.

So again, a line by its motion will 'generate' a surface - a moving surface generates a solid; and this surface is the locus of a line, and a solid the locus of a surface.

(Outlines of Geometry, or The Motion of a Point, by Walter M. Adams, B.A., 1866, p.13)

Locus

The only means recognized by the ancients for defining plane curves were (1) kinematic definitions in which a point moves subject to two superimposed motions, and (2) the section by a plane of a geometrical surface, such as a cone or sphere or cylinder. (A History of Mathematics, Carl B. Boyer, 1968, p. 209)

Pappus of Alexandria (c. 300 AD) composed the work Collection (Synagogue) in eight books. Book VII, Treasury of Analysis, contained works by Euclid, Apollonius, Aristaeus and Eratosthenes, which Pappus advised as most suitable for an advanced course in the method of analysis and synthesis (The Treasury of Mathematics: 1 Henrietta Midonick, 1965, p.401). This contained what is known as the Problem of Pappus, which is a generalization of 'the locus to three or four lines'. About 500 years earlier this locus to three or four lines was completely solved by Apollonius of Perga in Book III of his Conics. About 1300 years later, in 1637, René Descartes (1591-1661) solved the Problem of Pappus using his Methode, and in doing so, he was led to the invention of analytic geometry.

Some fundamental loci and useful hints

The locus of a point is the path traced by that point as it moves according to a given condition or conditions.

Locus problems can normally be solved by

(a) carefully choosing a set of coordinate axes, and

(b) using the theorems of coordinate geometry, for example, distance formulae, gradient formula, mid-point formula,   etc.

For example, if P moves so that its distance from one fixed point is equal to its distance from a second fixed point, then choose the origin as the midpoint of the two given points and choose the x-axis along the line joining the two given points.

Some fundamental results

1. The locus of a point which moves so that its distance from a fixed point is constant is a circle whose centre is the fixed point, and whose radius is the constant distance.

2. The locus of a point which moves so that its distance from a fixed straight line is constant is a pair of straight lines parallel to the given line, one on each side of it and at the given distance from it.

3. The locus of a point which moves so as always to be equidistant from two fixed points is the right bisector of the interval joining these points.

4. The locus of a point which moves so that it is always equidistant from two fixed intersecting straight lines is the pair of mutually perpendicular straight lines which bisect the angles between the fixed lines.

5. The locus of a point which moves so that a fixed interval subtends a constant angle at the point is the arc of a segment of a circle having the fixed interval as a chord.

6. The locus of a point which moves so that its distance from a fixed point is equal to its distance from a fixed line is a parabola whose focus is the fixed point and whose focal length is the perpendicular distance from the fixed point to the fixed line.

Practical tests for loci

The first five results above, and the following tests, can be found in the first edition of A Senior Geometry by P. Andersen & R. Hundt, (1946).

Straight-line loci

Select some particular position of the moving point, say . Now take any other position, say P. If we can show that  is either parallel to a fixed straight line or makes a constant angle with it, then the locus of P is a straight line or part thereof.

Circle loci

There are two tests:

(a) Show that the distance of the moving point from some fixed point is constant. The locus will then be a circle (or an arc of a circle) having the fixed point as centre and the constant distance as radius.

(b) Show that the angle subtended at the moving point by some fixed interval is constant. The point must then move on a circle (or arc thereof) having the fixed interval as a chord. 

Locus exercises

The following example can be found in standard high school textbooks:

Example 1

Find the locus of a point P which moves so that its distance from the line with equation x = 0 is equal to its distance from the line with equation y = 0.

So, the locus is the line with equation y = x.

However, this method gives only half the correct answer.

A better method

The locus is the pair of lines y = x, y = -x which are perpendicular to each other and which bisect the angles  

Follow-up exercise

Find the locus of a point P which moves so that its distance from the line x = 0 is twice its distance from the line y = 1.

Example 2

Find the locus of a point which moves so that the ratio of its distances from two fixed points A, B is equal to a constant k > 1, where AB = 2a units. Use k = 2 and a = 3. Also, describe and sketch the locus.

Take the line through A, B as the x-axis and the midpoint of AB as the origin of a set of rectangular coordinate axes.

Let P = (x, y) lie on the locus.

A = (3, 0) and B = (3, 0).

This reduces to  or  which is the equation of a circle of radius 4 units and centre (5, 0).

This circle is referred to as the Circle of Apollonius (of Perga).

Problem 1

1994 HSC 2/3 Unit Mathematics, Question 9(b)

The point P(x, y) is equidistant from the lines y = 3 and 3x + 4y -18 = 0 and lies in the shaded region of the diagram.

Find the equation of the locus of P.

Solution

Let P(x, y) lie on the locus.

Let M, N be the feet of the perpendiculars from P to y = 3 and 3x + 4y - 18 = 0.

 and

 and

Now PN = PM

 and

 This is the difference of two squares.

 or

These two lines are shown in the above diagram. It can be seen that only one of the two lines satisfies the given condition, and that the locus is

 x 2.

Problem 2

A student begins to climb a 4 metre ladder which is standing against a vertical wall. While the student's feet are on the rung which is 1 metre from the base, and his hands are on the rung 1 metre from the top of the ladder, the mechanism holding the ladder breaks and the ladder begins to slip (that is, the base moves horizontally away from the wall). The student keeps a firm grip on the ladder and the ladder continues to slip until the top reaches the ground.

(a) Find the locus of the student's hands, and the locus of his feet;

(b) Make a sketch of the loci;

(c) Describe the loci.

Solution

Hands

 
                                                            ... (1)

Feet

The locus is shown above. The locus is given by equations 1 & 2 above with x 0, y 0.

The student's hands and feet move along arcs of two ellipses.

Problem 3

Find the equation of the locus of a point P which moves so that the angle which is subtended at P by an interval AB of length 2 units is 45°. Sketch the locus and describe the locus.

Solution

 or

 or

These equations represent a circle, centre (0, 1) and radius =  units, and a circle, centre (0, -1) and radius =  units.

The above diagram represents the lines of force and the equipotentials of two long parallel circular-cylindrical wires of arbitrary diameters, placed an arbitrary distance apart, and charged as a condenser to a potential difference V.

The lines of force are circles of Apollonius and the equipotentials are circles of the type met in Problem 3.

Problem 4

This was the final question in the senior division paper of the 1997 Australian Mathematics Competition. Attempt this under competition conditions, that is, without the aid of a calculator.

In the diagram, angles LMN and LNM are 45 degrees. The intervals LM, LN, PQ and QR are each 1 unit long.

What is the farthest distance that R can be from the line MN? Select from options (A) to (E).

Solution

Choose rectangular coordinate axes as shown. Let R = (x, y). Let Q = (0, k). Then M = (-1, 0), N = (0, -1), P = (-x, 0). Let RS be perpendicular to MN.

Now, in    So

Now Q is the midpoint of PR.

So

So  or

The positive sign will be used below. So the equation of the locus of R is  and this represents a quarter-ellipse (with the point (0, 2) omitted) if P is constrained to move along the interval LM, and Q moves along the non-negative y-axis.

Now the equation of MN is

Now the maximum value of RS will occur where  that is,

So  giving  or  and

The corresponding value of RS is

Now  which is negative for

Hence the maximum value of RS is  units, which is equivalent to option (E).

Related graphs

Closely related to the idea of a locus are the following curves:

Some of them can be investigated through the Board of Studies NSW home page in the Mathematical Curves section, using the address: http://www.boardofstudies.nsw.edu.au.

There are 60 famous curves and related curves to investigate.