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.
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 h^{2} = (9 cosα )^{2} h^{2} = 81 cos^{2}α h^{2}/81 = cos^{2}α -----(1) |
k = 9 sinα k^{2} = (9 sinα)^{2} k^{2} = 81 sin^{2}α k^{2}/81 = sin^{2}α -----(2) |
(1) + (2)
cos^{2}α + sin^{2}α = (h^{2}/81) + (k^{2}/81)
1 = (h^{2 }+ k^{2})/81
81 = h^{2}+ k^{2}
By replacing h = x and k = y, we get
x^{2} + y^{2} = 81
(ii) (9 cosα , 6 sinα)
h = 9 cosα h^{2} = (9 cosα )^{2} h^{2} = 81 cos^{2}α h^{2}/81 = cos^{2}α -----(1) |
k = 6 sinα k^{2} = (6 sinα)^{2} k^{2} = 36 sin^{2}α k^{2}/36 = sin^{2}α -----(2) |
(1) + (2)
cos^{2}α + sin^{2}α = (h^{2}/81) + (k^{2}/36)
1 = (h^{2}/81) + (k^{2}/36)
By replacing h = x and k = y, we get
(x^{2}/81) + (y^{2}/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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