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
From Worked-out 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.
May 24, 24 06:42 PM
May 24, 24 06:23 PM
May 24, 24 06:22 PM
May 24, 24 05:37 PM
May 24, 24 05:09 PM
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.