HOW TO FIND THE LOCUS OF A POINT

About "How to Find the Locus of a Point"

How to Find the Locus of a Point ?

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, say (h, k) to P

(ii) Express the given conditions as equations in terms of the known quantities and unknown parameters.

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

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

Define locus :

The path traced out by a moving point under certain conditions is called the locus of that point. Alternatively, when a point moves in accordance with a geometrical law, its path is called locus. The plural of locus is loci.

How to Find the Locus of a Point - Practice questions

Question 1 :

Find the locus of P, if for all values of α, the co-ordinates of a moving point P is

(i) (9cosα , 9 sinα)

Solution :

h  =  9 cosα 

h2  =  (9 cosα )2

h2  =  81 cos2α

h2/81  =  cos2α  -----(1)

k  =  9 sinα

k2  =  (9 sinα)2

k2  =  81 sin2α

k2/81  =   sin2α -----(2)

(1) + (2)  

cos2α + sin2α  =  (h2/81) + (k2/81)

1  =  (h2 k2)/81

81  =  h2k2

By replacing h = x and k = y, we get

x2 + y2  =  81

(ii) (9 cosα , 6 sinα)

h  =  9 cosα 

h2  =  (9 cosα )2

h2  =  81 cos2α

h2/81  =  cos2α  -----(1)

k  =  6 sinα

k2  =  (6 sinα)2

k2  =  36 sin2α

k2/36  =   sin2α -----(2)

(1) + (2)  

cos2α + sin2α  =  (h2/81) + (k2/36)

1  =  (h2/81) + (k2/36)

By replacing h = x and k = y, we get

(x2/81) + (y2/36)  =   1

Question 2 :

Find the locus of a point P that moves at a constant distant of (i) two units from the x-axis (ii) three units from the y-axis.

Solution :

(i) two units from the x-axis

Equation of locus : y = 2

(ii) three units from the y-axis.

Equation of locus : x = 3

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