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
For locus problems with strings:
\begin{itemize}
\item A taut string enforces a fixed distance,
\item Multiple constraints form envelopes of curves,
\item Symmetry helps identify the correct shape.
\end{itemize}
Step 1: Let the fixed points be \(1\) and \(2\), and the lengths of strings \(P\) and \(Q\) be equal to \(L\).
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Step 2: If at every instant \emph{one} of the two strings is fully stretched (taut), then the red dot always lies at a point such that:
\begin{itemize}
\item Its distance from point \(1\) is \(\le L\),
\item Its distance from point \(2\) is \(\le L\),
\item At least one of these distances is exactly \(L\).
\end{itemize}
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Step 3: The locus of points at a fixed distance \(L\) from point \(1\) is a circle with center \(1\).
Similarly, for point \(2\), it is a circle with center \(2\).
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Step 4: The path traced by the dot is the outer envelope of the two circles of radius \(L\) centered at points \(1\) and \(2\). This envelope forms a smooth, lens-shaped curve.
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Step 5: Among the options, only Option D represents this lens-shaped locus.
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Final Answer:
\[
\boxed{D}
\]
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