Key points to solve the problem:

• Idea of distance formula- Distance between two points A(x₁,y₁) and B(x₂,y₂) is given by- AB =

• Idea of section formula- Let two points A(x₁,y₁) and B(x₂,y₂) forms a line segment. If a point C(x,y) divides line segment AB in the ratio of m:n internally, then coordinates of C is given as:

C =

How to approach: To find the locus of a point we first assume the coordinate of the point to be (h, k) and write a mathematical equation as per the conditions mentioned in the question and finally replace (h, k) with (x, y) to get the locus of the point.

Let the coordinates of a point whose locus is to be determined to be (h, k). Name the moving point to be C

Assume the two perpendicular lines on which rod slides are x and y-axis respectively.

Here line segment AB represents the rod of length l also ΔADB formed is a right triangle. Coordinates of A and B are assumed to be (0,b) and (a,0) respectively.

∴ a² + b² = l²…eqn 1

As, (h,k) divides AB in ratio of 1:2

∴ from section formula we have coordinate of point C as-

C = =

As, a and b are assumed parameters so we have to remove it.

∵ h = 2a/3 ⇒ a = 3h/2

And k = b/3 ⇒ b = 3k

From eqn 1:

a² + b² = l²

∴

⇒

Replace (h,k) with (x,y)

Thus, the locus of a point on the rod is: ….ans