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NDA 2024 · PYQ · Coordinate Geometry / Distance on Line · medium

Two points P and Q lie on line y = 2x + 3. These two points P and Q are at a distance 2 units from another point R(1, 5). What are the coordinates of the points P and Q?

  1. A.(1 + 2/√5, 5 + 4/√5), (1 – 2/√5, 5 – 4/√5)✓ Correct
  2. B.(3 + 2/√5, 5 + 4/√5), (–1 – 2/√5, 5 – 4/√5)
  3. C.(1 – 2/√5, 5 + 4/√5), (1 + 2/√5, 5 – 4/√5)
  4. D.(3 – 2/√5, 5 + 4/√5), (–1 + 2/√5, 5 – 4/√5)

Explanation

Note R(1, 5) lies on y = 2x + 3 since 2(1) + 3 = 5. The direction along the line (slope 2) has unit vector (1, 2)/√5. Points at distance 2 from R along the line: (1 ± 2/√5, 5 ± 4/√5). So P = (1 + 2/√5, 5 + 4/√5) and Q = (1 – 2/√5, 5 – 4/√5).
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