The ends of a rod of length L move on two mutually perpendicular lines. Find the locus of the point on the road, which divides it in the ratio 2:1.

To find the locus of a point on a rod of length L that divides it in the ratio 2:1, where the ends of the rod move along two mutually perpendicular lines, we can approach this problem step by step. Let’s break it down clearly.

Setting Up the Problem

Imagine we have a rod of length L, with one end moving along the x-axis and the other end moving along the y-axis. We can denote the position of the ends of the rod as follows:

• Let the position of the first end (on the x-axis) be at point A(x, 0).
• Let the position of the second end (on the y-axis) be at point B(0, y).

Finding the Coordinates of the Dividing Point

We need to find the coordinates of the point P that divides the rod AB in the ratio 2:1. According to the section formula in coordinate geometry, the coordinates of point P can be calculated as:

P = (mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)

In our case, m = 2, n = 1, and the coordinates of A and B are (x, 0) and (0, y) respectively. Plugging these values into the formula gives us:

P = ((2*0 + 1*x)/(2+1), (2*y + 1*0)/(2+1))

This simplifies to:

P = (x/3, 2y/3)

Using the Length of the Rod

Since the length of the rod is constant (L), we have the relationship:

AB = √[(x - 0)² + (0 - y)²] = L

Squaring both sides gives us:

x² + y² = L²

Expressing the Locus of Point P

Now, we need to express the locus of point P in terms of x and y. We already found that the coordinates of P are (x/3, 2y/3). To find the relationship between these coordinates, we can express x and y in terms of P's coordinates:

• If we let P = (X, Y), then we have:
• X = x/3 → x = 3X
• Y = 2y/3 → y = (3/2)Y

Substituting these into the equation of the rod's length:

(3X)² + ((3/2)Y)² = L²

This simplifies to:

9X² + (9/4)Y² = L²

Final Form of the Locus

To express this in a standard form, we can divide the entire equation by L²:

(9X²/L²) + (9Y²/(4L²)) = 1

Rearranging gives us:

(X²/(L²/9)) + (Y²/(4L²/9)) = 1

This is the equation of an ellipse centered at the origin, with semi-major axis L/3 along the x-axis and semi-minor axis (2L/3) along the y-axis.

Conclusion

Thus, the locus of the point that divides the rod in the ratio 2:1, as the ends of the rod move along two perpendicular lines, is an ellipse. This geometric interpretation helps visualize how the point P moves as the ends of the rod shift along their respective axes.