Locus : Co-ordinate Geometry

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 :

(i) If we are finding the equation of the locus of a point P, assign coordinates (h, k) to P

(ii) Express the given conditions as equations in terms of the known quantities to facilitate calculations. We sometimes include some unknown quantities known as parameters.

(iii) Eliminate the parameters, so that the eliminate contains only h, k and known quantities.

(iv) Replace h by x, and k by y, in the eliminate. The resulting equation would be the equation of the locus of P.

(v) 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.

Illustration : Find the locus of the middle points of the segment of a line passing through the point of intersection of the lines ax + by + c = 0 and lx + my + n = 0 and intercepted between the axes.

Solution: Any line (say L = 0) passing through the point of intersection of ax + by + c = 0 and lx + my + n = 0 is (ax+ by + c) + λ(lx + my + n)=0, where λ is any real number.

Point of intersection of L = 0 with axes are

$ \displaystyle (-\frac{c+\lambda n}{a + \lambda l} , 0 ) \; and \; ( 0 , -\frac{c+\lambda n}{b+ \lambda m} ) $

Let the mid point be (h, k).

Then

$ \displaystyle h = -\frac{1}{2}(\frac{c+\lambda n}{a + \lambda l}) $

and

$ \displaystyle k = -\frac{1}{2}(\frac{c+\lambda n}{b + \lambda m}) $

Eliminating λ, we get

$ \displaystyle \frac{2 a h + c}{2 h l + n} = \frac{2 k b + c}{2 k m + n} $

The required locus is: 2(am − lb)xy = (lc − an)x + (nb − mc)y

Exercise :

(i) Find the locus of the moving point P such that 2PA=3PB where A is (0, 0) and B is (4, −3).

(ii) The co-ordinates of three points O, A, B are (0, 0), (0, 4) and (6, 0) respectively. A point P moves so that the area of the triangle POA is always, twice the area of the triangle POB. Find the locus of P

(iii) 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

(iv) The ends of a rod of length l move on two mutually perpendicular lines. Find the locus of the point on the rod, which divides it in the ratio 2 : 1

(v) Two points P(a, 0) and Q(−a , 0) are given and R is a variable point on one side of the line PQ such that ∠RPQ − ∠ RQP is a positive constant 2α. Show that the locus of R is x2 − y2 ± 2xy cot2α − a2 = 0

Also Read :

Co-ordinate Geometry
Area of a Triangle
Equations of Straight Line in Different Forms
Angle between Two Straight Lines
Bisectors of the angles b/w two lines
Equation of reflected ray
Family of Lines , Concurrency of Straight Lines
PAIR OF STRAIGHT LINES
Solved Problems : Straight Lines

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