TS EAMCET · Maths · Sets and Relations

Let \(S\) be the set of points on \(X\)-axis lying at a distance of \(d\) units from \((3,4)\). Which of the following is true?

A \(S\) is an empty set if \(d < 4\)
B \(S\) contains infinitely many points if \(d < 4\)
C \(S\) contains at least two points if \(d=4\)
D \(S\) contains exactly three points for any \(d>4\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(S\) is an empty set if \(d < 4\)

Step-by-step Solution

Detailed explanation

Let \(S=(x, 0)\) and \(P=(3,4)\) \(\therefore \quad d^2=S P^2=(x-3)^2+16 \Rightarrow(x-3)^2=d^2-16\) If \(d \in(-4,4)\), then \(\left(+d^2-16 < 0\right)\) and then \((x-3)^2\) becomes negative, which is impossible. \(\Rightarrow \quad S\) is an empty set if \(d < 4\).

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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