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 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 = h2+ k2
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
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
©All rights reserved. onlinemath4all.com
Aug 31, 24 08:51 PM
Aug 31, 24 08:47 PM
Aug 31, 24 06:59 PM