Q. 103.5( 2 Votes )

# A rod of length l slides between the two perpendicular lines. Find the locus of the point on the rod which divides it in the ratio 1 : 2.

Answer :

**Key points to solve the problem:**

• **Idea of distance formula-** Distance between two points A(x_{1},y_{1}) and B(x_{2},y_{2}) is given by- **AB =**

• **Idea of section formula-** Let two points A(x_{1},y_{1}) and B(x_{2},y_{2}) 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.

∴ a^{2} + b^{2} = l^{2}**…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:

a^{2} + b^{2} = l^{2}

∴

⇒

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

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

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