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Foot of the perpendicular from a point to a line

This type of question, asked 9 times: 27 Jun 2022, 25 Jan 2023, 13 Apr 2023, 30 Jan 2024, 6 Apr 2024, 22 Jan 2025, 29 Jan 2025, 21 Jan 2026 S1, 2 Apr 2026 S2. All 9 questions below, oldest first.

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  1. Question 1

    27 Jun 2022

    163
    Let the foot of the perpendicular from the point (1,2,4)(1,2,4) on the line x+24=y−12=z+13\frac{x + 2}{4}=\frac{y - 1}{2}=\frac{z + 1}{3} be PP. Then the distance of PP from the plane 3x+4y+12z+23=03x + 4y + 12z + 23 = 0

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  2. Question 2

    25 Jan 2023

    153
    The foot of perpendicular of the point (2,0,5)(2,0,5) on the line x+12=y−15=z+1−1\frac{x + 1}{2}=\frac{y - 1}{5}=\frac{z + 1}{- 1} is (α,β,γ)(\alpha,\beta,\gamma). Then, which of the following is NOT correct?

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  3. Question 3

    13 Apr 2023

    158
    Let NN be the foot of perpendicular from the point PP (1,−2,3)(1, - 2,3) on the line passing through the points (4,5,8)(4,5,8) and (1,−7,5)(1, - 7,5). Then the distance of NN from the plane 2x−2y+z+5=02x - 2y + z + 5 = 0 is

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  4. Question 4

    30 Jan 2024

    70
    Let (α,β,γ)(\alpha,\beta,\gamma) be the foot of perpendicular from the point (1,2,3)(1,2,3) on the line x+35=y−12=z+43\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}. Then 19(α+β+γ)19(\alpha+\beta+\gamma) is equal to:

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  5. Question 5

    6 Apr 2024

    89
    Let PP be the point (10,−2,−1)(10, - 2, - 1) and QQ be the foot of the perpendicular drawn from the point R(1,7,6)R(1,7,6) on the line passing through the points (2,−5,11)(2, - 5,11) and (−6,7,−5)( - 6,7, - 5). Then the length of the line segment PQPQ is equal to

  6. Question 6

    22 Jan 2025

    74
    Let L1:x−13=y−1−1=z+10L_{1}:\frac{x - 1}{3}=\frac{y - 1}{- 1}=\frac{z + 1}{0} and L2:x−22=y0=z+4α,α∈RL_{2}:\frac{x - 2}{2}=\frac{y}{0}=\frac{z + 4}{\alpha},\alpha \in R, be two lines, which intersect at the point BB. If PP is the foot of perpendicular from the point A(1,1,−1)A(1,1, - 1) on L2L_{2}, then the value of 26α( PB)226\alpha(\text{ }PB)^{2} is _____\_\_\_\_\_ .

  7. Question 7

    29 Jan 2025

    138
    Let PP be the foot of the perpendicular from the point (1,2,2)(1,2,2) on the line L:x−11=y+1−1=z−22L:\frac{x - 1}{1}=\frac{y + 1}{- 1}=\frac{z - 2}{2}. Let the line r→=(−i^+j^−2k^)+λ(i^−j^+k^),λ∈R\overrightarrow{r}= ( -\widehat{i}+\widehat{j}- 2\widehat{k}) + \lambda(\widehat{i}-\widehat{j}+\widehat{k}),\lambda \in R, intersect the line L at Q . Then 2(PQ)22(PQ)^{2} is equal to:

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  8. Question 8

    21 Jan 2026 S1

    64
    Let (α,β,γ)(\alpha,\beta,\gamma) be the co-ordinates of the foot of the perpendicular drawn from the point (5,4,2)(5,4,2) on the line r→=(−i^+3j^+k^)+λ(2i^+3j^−k^)\overrightarrow{r}= ( -\widehat{i}+ 3\widehat{j}+\widehat{k}) + \lambda(2\widehat{i}+ 3\widehat{j}-\widehat{k}). Then the length of the projection of the vector αi^+βj^+γk^\alpha\widehat{i}+ \beta\widehat{j}+ \gamma\widehat{k} on the vector 6i^+2j^+3k^6\widehat{i}+ 2\widehat{j}+ 3\widehat{k} is :

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  9. Question 9

    2 Apr 2026 S2

    65
    Let the point A be the foot of perpendicular drawn from the point P(a,b,0)P(a,b,0) on the line x−12=y−21=z−α3\frac{x- 1}{2}=\frac{y- 2}{1}=\frac{z-\alpha}{3}. If the midpoint of the line segment PA is (0,34,−14)\left( 0,\frac{3}{4},\frac{- 1}{4} \right), then the value of a2+b2+α2a^{2}+b^{2}+\alpha^{2} is equal to :

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