To solve the worked-out 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.
Worked-out Problems on Locus of a Moving Point:
1. The sum of the intercept cut off from the axes of co-ordinates 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 co-ordinates in the ratio 2 : 3.
Solution:
Let us assume that the variable straight line at any position intersects the x-axis at A (a, 0) and the y-axis at B (0, b).
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 co-ordinates 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 co-ordinates 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 co-ordinates 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 co-ordinates 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
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