PYQ Vault
EbooksTutor

Two points on a line at the same distance from a given point (use the foot of the perpendicular)

This type of question, asked 5 times: 28 Jun 2022, 26 Jul 2022, 1 Feb 2024, 4 Apr 2025, 5 Apr 2026 S2. All 5 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    28 Jun 2022

    75
    If two distinct points Q,RQ,R lie on the line of intersection of the planes −x+2y−z=0-x + 2y - z = 0 and 3x−5y+2z=03x - 5y + 2z = 0 and PQ=PR=18PQ = PR =\sqrt{18} where the point PP is (1,−2,3)(1, - 2,3), then the area of the triangle PQRPQR is equal to

    Tap an option to check your answer.

  2. Question 2

    26 Jul 2022

    90
    Let QQ and RR be two points on the line x+12=y+23=z−12\frac{x + 1}{2}=\frac{y + 2}{3}=\frac{z - 1}{2} at a distance 26\sqrt{26} from the point P(4,2,7)P(4,2,7). Then the square of the area of the triangle PQRPQR is

  3. Question 3

    1 Feb 2024

    158
    Let PP and QQ be the points on the line x+38=y−42=z+12\frac{x + 3}{8}=\frac{y - 4}{2}=\frac{z + 1}{2} which are at a distance of 6 units from the point R(1,2,3)R(1,2,3). If the centroid of the triangle PQRPQR is (α,β,γ)(\alpha,\beta,\gamma), then α2+β2+γ2\alpha^{2}+\beta^{2}+\gamma^{2} is:

    Tap an option to check your answer.

  4. Question 4

    4 Apr 2025

    59
    Let A and B be two distinct points on the line L:x−63=y−72=z−7−2L:\frac{x - 6}{3}=\frac{y - 7}{2}=\frac{z - 7}{- 2}. Both A and B are at a distance 2172\sqrt{17} from the foot of perpendicular drawn from the point (1,2,3)(1,2,3) on the line L . If O is the origin, then OA→⋅OB→\overrightarrow{OA}\cdot\overrightarrow{OB} is equal to:

    Tap an option to check your answer.

  5. Question 5

    5 Apr 2026 S2

    64
    Let a triangle PQR be such that P and Q lie on the line x+38=y−42=z+12\frac{x+ 3}{8}=\frac{y- 4}{2}=\frac{z+ 1}{2} and are at a distance of 6 units from R(1,2,3)R(1,2,3). If (α,β,γ)(\alpha,\beta,\gamma) is the centroid of ΔPQR\Delta PQR, then α+β+γ\alpha+\beta+\gamma is equal to:

    Tap an option to check your answer.