A red dot is connected with two non-elastic strings P and Q of equal length. The strings are fixed at points 1 and 2, as shown below. Consider both the strings, points 1, 2, and the red dot are all on the same plane throughout the operations. If one of the strings is fully stretched (taut) at all times, what will be the shape of the path traced by the dot?
Show Hint
The boundary of the intersection of two circles always has sharp corners at the points of intersection. Any smooth curve option (like an ellipse) can be eliminated instantly.
Step 1: Understanding the Question:
This question belongs to the topic of
Geometric Loci and Kinematics.
We are asked to find the path (locus) traced by a point (the red dot) connected to two fixed points (1 and 2) via two non-elastic strings P and Q of equal length \(L\). One of the strings is always fully stretched.
Step 2: Key Formula or Approach:
Let \(r_1\) be the distance from the dot to Point 1, and \(r_2\) be the distance from the dot to Point 2.
• Since the strings have a maximum length \(L\), we must have \(r_1 \le L\) and \(r_2 \le L\) at all times.
• The condition "one of the strings is fully stretched at all times" means that at any point on the path, either \(r_1 = L\) OR \(r_2 = L\).
• Therefore, the path of the dot is the boundary of the intersection of two circular disks of radius \(L\) centered at Point 1 and Point 2.
Step 3: Detailed Explanation:
• Let \(C_1\) be the circle of radius \(L\) centered at Point 1, and \(C_2\) be the circle of radius \(L\) centered at Point 2.
• When string P is fully stretched (\(r_1 = L\)), the dot moves along the circle \(C_1\). However, it is constrained by string Q, so it can only move on the arc of \(C_1\) where \(r_2 \le L\).
• When string Q is fully stretched (\(r_2 = L\)), the dot moves along the circle \(C_2\), constrained such that \(r_1 \le L\).
• The intersection of these two constraints forms a lens-shaped region (known as a vesica piscis).
• The path traced by the dot when at least one string is taut is the outer boundary of this lens.
• Because the boundary is made of two circular arcs meeting at the intersection points of the two circles, the path must have sharp, pointed corners at the top and bottom.
• Let us analyze the options:
- A: Shows an open hyperbola/X-shape.
- B: Shows a smooth ellipse, which lacks the sharp corners.
- C: Shows a figure-8 shape.
- D: Shows a closed, lens-like shape with sharp pointed vertices at the top and bottom. This is the correct vesica piscis boundary.
Step 4: Final Answer:
Option (D) correctly shows the path traced by the red dot.
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