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.
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:
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
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