Locus of a point
It is important to
understand that a point is not a thing, but a place. We indicate the position of a point by placing a dot. In plane
analytical geometry, points are defined as ordered pairs of real numbers, say,
(x, y) with reference to the coordinate system.
horizontal line is called the x-axis; and the line
vertical to the x-axis is called the y-axis. Intersection of these two axes is called the origin. Any
point P in the plane can be located by a unique ordered pair
of numbers(x, y) where x gives the distance
between the point P and the y-axis and y denote the distance
between the point P and the x-axis. Note that if x is negative it lies
left of y − axis, similarly if y is negative it lies
below the x-axis. In applications, often letters other than x and y are used, and
different scales are chosen in the horizontal and vertical directions.
The following illustrations shows
some cases of loci and its different uses.
6.1: In cricket,
when a ball is bowled by a bowler, the path traced out by the ball is the locus of the ball. Whenever there is dispute between
batsmen and the fielders for leg before wicket (LBW) decisions, the locus of
the ball solves the crises, raised by the players for review, through the third
umpire. The likely path of the ball can be projected forward, through the
batsman’s legs, to see whether it would have hit the stumps or not.
Consultation of the third umpire, for conventional slow motion or Hawk Eye, the
probable decision will be taken. This method is currently sanctioned in
6.2: Suppose P be a point on the rim (circumference) of a circular
wheel. When the circle is rolling without
slipping along a straight line, the locus of the point P on the rim is shown in figure.
The path traced out by the point P is known as cycloid. (Try
yourself by taking a point inside the circle.
6.3: A missile is launched from the army ship to attack
and another from the land to intercept it. The loci of the
missiles are shown in figure.
Locus of missiles play a vital
role in many wars. During the Gulf War (2 Aug1990-28 Feb 1991), Iraq attacked
Israeli cities with Scud missiles. To defend from Scud attack, Israel used
Patriot missiles to shoot down enemy missiles. To launch a satellite or space
shuttle successfully, the determination of path plays an crucial role in space
An equation in the two variables
x and y will ordinarily be satisfied by infinitely many pair of real value of x and y. Every such pair is called a
real solution of the equation. Each real solution of the equation will have its
graph. The collection of all these graphs is called the locus of the given
The following table shows some
important loci in mathematics
Now let us discuss the ways of
finding the locus of the points. The equation of the locus is the relation that
exists between the coordinates of all the points strictly lying on the path.
If we are finding the equation of the locus of a point P, assign
coordinates, say (h, k) to P
Express the given conditions as equations in terms of the known
quantities and unknown parameters.
Eliminate the parameters, so that the resulting equation
contains only h, k and known quantities.
Replace h by x, and k by y, in the resulting equation. The resulting equation is the
equation of the locus of point P.