Locus
When a point moves in a plane under
certain geometrical conditions, the point traces out a path. This path of a
moving point is called its locus.
Equation of Locus:
The equation to a locus
is the relation which exists between the coordinates of any point on the path,
and which holds for no other point except those lying on the path.
Procedure for finding the
equation of the locus of a point
- If we are finding the equation of the
locus of a point P, assign coordinates (h, k) to P.
- Express the given conditions as
equations in terms of the known quantities to facilitate calculations, We
sometimes include some unknown quantities known as parameters.
- Eliminate the parameters, so that the
eliminant contains only h, k and known quantities.
- Replace h by x, and k by y, in the
eliminant. The resulting equation would be the equation of the locus of P.
- If x and y coordinates of the moving
point are obtained in terms of a third variable t (called the parameter),
eliminate t to obtain the relation in x and y and simplify this relation. This
will give the required equation of locus.
Example 1
The
ends of a rod of length 
move on two
mutually perpendicular lines. Find the locus of the point on the rod, which
divides it in the ratio 2 : 1.
Solution:
|
Suppose
the two perpendicular lines are x = 0 and y = 0 and the rod intercepts a and
b cuts on these two lines respectively, then the two points on these lines
are (0, a) and (b, 0). The point P has coordinates given by h =  , k = 
|
|
Also 
=
a2 + b2
Thus the required locus is x2+
.
Example 2
Find
the locus of a point which moves so that the sum of its distances from (3, 0)
and (–3, 0) is less than 9.
Solution:
Let P(h, k)be the moving point such
that the sum of its distances from A (3, 0) and B (–3, 0) is less than 9. .
Then, PA + PB < 9
&⇒
&⇒
&⇒
(h-3)2 + k2 < 81 + (h+3)2 + k2 –
18 
&⇒
–12h – 81 < –18 
&⇒
4h + 27 > 6
&⇒
(4h + 27)2 > 36 [(h+3)2 + k2]
&⇒
20h2 + 36k2 < 405
Hence the locus of (h, k) is
20x2 + 36y2
< 405.
