# Worked-out Problems on Locus of a Moving Point

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). clearly,

AB is the part of the line intercepted between the co-ordinates axes. Further assume that the point (h, k) divides the line-segment 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 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

x2/a2 + y2/b2 = cos2 θ + sin2 θ

or, x2/a2 + y2/b2 = 1

Which 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