To solve the workedout problems on locus of a moving point we need to follow the method of obtaining the equation of the locus. Recall and consider the steps to find the equation to the locus of a moving point.
Workedout Problems on Locus of a Moving Point:
1. The sum of the intercept cut off from the axes of coordinates by a variable straight line is 10 units. Find the locus of the point which divides internally the part of the straight line intercepted between the axes of coordinates in the ratio 2 : 3.
Solution: Let us assume that the variable straight line at any position intersects the xaxis at A (a, 0) and the yaxis at B (0, b). clearly,
AB is the part of the line intercepted between the coordinates axes. Further assume that the point (h, k) divides the linesegment AB internally in the ratio 2 : 3. Then we have,H = (2 · 0 + 3 · a)/(2 + 3)
or, 3a = 5h
or, a = 5h/3
And k = (2 · b + 3 · a)/(2 + 3)
or, 2b = 5k
or, b = 5k/2
Now, by problem,
A + b = 10
or, 5h/3 + 5k/2 = 10
or, 2h + 3k = 12
Therefore, the required equation to the locus of (h, k) is 2x + 3y = 12.
2. For all value of the coordinates of a moving point P are (a cos θ, b sin θ); find the equation to the locus of P.
Solution: Let (x, y) be the coordinates of any point on the locus traced out by the moving point P. then we shall have ,
x = a cos θ
or, x/a = cos θ
and y = b sin θ
or, y/b = sin θ
x^{2}/a^{2} + y^{2}/b^{2} = cos^{2} θ + sin^{2} θWhich is the required equation to the locus of P.
3. The coordinates of any position of a moving point P are given by {(7t – 2)/(3t + 2)}, {(4t + 5)/(t – 1)}, where t is a variable parameter. Find the equation to the locus of P.
Solution: Let (x, y) be the coordinates of any point on the locus traced out by the moving point P. then, we shall have,
x = (7t – 2)/(3t + 2)
or, 7t – 2 = 3tx + 2x
or, t(7 – 3x) = 2x + 2
or, t = 2(x + 1)/(7 – 3x) …………………………. (1)
And
y = (4t + 5)/(t – 1)
or, yt – y = 4t + 5
Or, t (y – 4) = y +5
or , t = (y + 5)/(y – 4)………………………….. (2)
From (1) and (2) we get,
(2x + 2)/(7 – 3x) = (y + 5)/( y – 4)
or, 2xy  8x + 2y – 8 = 7y – 3xy + 35 – 15x
or, 5xy + 7x 5y = 43, which is the required education to the locus of the moving point P.
● Locus
From Workedout Problems on Locus of a Moving Point to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.