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Mirror image of a point in a line

This type of question, asked 19 times: 2021 Paper 2, 2021 Paper 12, 28 Jun 2022, 27 Jul 2022, 27 Jan 2024, 31 Jan 2024, 1 Feb 2024, 4 Apr 2024, 5 Apr 2024, 8 Apr 2024, 9 Apr 2024, 28 Jan 2025, 2 Apr 2025, 22 Jan 2026 S1, 23 Jan 2026 S2, 28 Jan 2026 S2, 4 Apr 2026 S2, 5 Apr 2026 S2, 6 Apr 2026 S1. All 19 questions below, oldest first.

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  1. Question 1

    2021 Paper 2

    61
    Let a,b∈Ra,b\in R. If the mirror image of the point P(a,6,9)P(a,6,9) with respect to the line x−37=y−25=z−1−9\frac{x- 3}{7}=\frac{y- 2}{5}=\frac{z- 1}{- 9} is (20,b,−a−9)(20,b, -a- 9), then ∣a+b∣|a+b| is equal to:

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  2. Question 2

    2021 Paper 12

    52
    Consider the line LL given by the equation x−32=y−11=z−21\frac{x - 3}{2}=\frac{y - 1}{1}=\frac{z - 2}{1}. Let QQ be the mirror image of the point (2,3,−1)(2,3, - 1) with respect to LL. Let a plane PP be such that it passes through QQ, and the line LL is perpendicular to PP. Then which of the following points is on the plane PP?

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  3. Question 3

    28 Jun 2022

    171
    Let the image of the point P(1,2,3)P(1,2,3) in the line L:x−63=y−12=z−23L:\frac{x - 6}{3}=\frac{y - 1}{2}=\frac{z - 2}{3} be QQ. let R(α,β,γ)R(\alpha,\beta,\gamma) be a point that divides internally the line segment PQPQ in the ratio 1:31:3. Then the value of 22(α+β+γ)22(\alpha+\beta+\gamma) is equal to

  4. Question 4

    27 Jul 2022

    164
    If the length of the perpendicular drawn from the point P(a,4,2),a>0P(a,4,2),a > 0 on the line x+12=y−33=z−1−1\frac{x + 1}{2}=\frac{y - 3}{3}=\frac{z - 1}{- 1} is  26\ 2\sqrt{6} units and Q(α1,α2,α3)Q\left( \alpha_{1},\alpha_{2},\alpha_{3} \right) is the image of the point PP in this line, then a+∑i=13αia+\sum_{i= 1}^{3} \alpha_{i} is equal to:

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  5. Question 5

    27 Jan 2024

    153
    Let the image of the point (1,0,7)(1,0,7) in the line x1=y−12=z−23\frac{x}{1}=\frac{y - 1}{2}=\frac{z - 2}{3} be the point (α,β,γ)(\alpha,\beta,\gamma). Then which one of the following points lies on the line passing through (α,β,γ)(\alpha,\beta,\gamma) and making angles 2π3\frac{2\pi}{3} and 3π4\frac{3\pi}{4} with y-axis and z-axis respectively and an acute angle with xx-axis?

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  6. Question 6

    31 Jan 2024

    156
    Let (α,β,γ)(\alpha,\beta,\gamma) be mirror image of the point (2,3,5)(2,3,5) in the line x−12=y−23=z−34\frac{x - 1}{2}=\frac{y - 2}{3}=\frac{z - 3}{4}. Then 2α+3β+4γ2\alpha + 3\beta + 4\gamma is equal to

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  7. Question 7

    1 Feb 2024

    165
    If the mirror image of the point P(3,4,9)P(3,4,9) in the line x−13=y+12=z−21\frac{x - 1}{3}=\frac{y + 1}{2}=\frac{z - 2}{1} is (α,β,γ)(\alpha,\beta,\gamma), then 14(α+β+γ)14(\alpha + \beta + \gamma) is :

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  8. Question 8

    4 Apr 2024

    179
    Consider a line LL passing through the points P(1,2,1)P(1,2,1) and Q(2,1,−1)Q(2,1, - 1). If the mirror image of the point A(2,2,2)A(2,2,2) in the line LL is (α,β,γ)(\alpha,\beta,\gamma), then α+β+6γ\alpha+\beta+ 6\gamma is equal to ....

  9. Question 9

    5 Apr 2024

    159
    Let (α,β,γ)(\alpha,\beta,\gamma) be the image of the point (8,5,7)(8,5,7) in the line x−12=y+13=z−25\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 2}{5}. Then α+β+γ\alpha + \beta + \gamma is equal to

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  10. Question 10

    8 Apr 2024

    174
    Let P(α,β,γ)P(\alpha,\beta,\gamma) be the image of the point Q(1,6,4)Q(1,6,4) in the line x1=y−12=z−23\frac{x}{1}=\frac{y- 1}{2}=\frac{z- 2}{3}. Then 2α+β+γ2\alpha+\beta+\gamma is equal to

  11. Question 11

    9 Apr 2024

    177
    The square of the distance of the image of the point (6,1,5)(6,1,5) in the line x−13=y2=z−24\frac{x- 1}{3}=\frac{y}{2}=\frac{z- 2}{4}, from the origin is

  12. Question 12

    28 Jan 2025

    62
    If the image of the point (4,4,3)(4,4,3) in the line x−12=y−21=z−13\frac{x - 1}{2}=\frac{y - 2}{1}=\frac{z - 1}{3} is (α,β,γ)(\alpha,\beta,\gamma), then α+β+γ\alpha + \beta + \gamma is equal to

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  13. Question 13

    2 Apr 2025

    126
    If the image of the point P(1,0,3)P(1,0,3) in the line joining the points A(4,7,1)A(4,7,1) and B(3,5,3)B(3,5,3) is Q(α,β,γ)Q(\alpha,\beta,\gamma), then α+β+γ\alpha+\beta+\gamma is equal to.

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  14. Question 14

    22 Jan 2026 S1

    61
    If the image of the point P(1,2,a)P(1,2,a) in the line x−63=y−72=7−z2\frac{x - 6}{3}=\frac{y - 7}{2}=\frac{7 - z}{2} is Q(5, b,c)Q(5,\text{ }b,c), then a2+b2+c2a^{2}+b^{2}+c^{2} is equal to

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  15. Question 15

    23 Jan 2026 S2

    72
    If the image of the point P(a,2,a)P(a,2,a) in the line x2=y+a1=z1\frac{x}{2} = \frac{y + a}{1} = \frac{z}{1} is Q and the image of Q in the line x−2b2=y−a1=z+2b−5\frac{x - 2b}{2} = \frac{y - a}{1} = \frac{z + 2b}{- 5} is PP, then a+ba + b is equal to ____\_\_\_\_

  16. Question 16

    28 Jan 2026 S2

    66
    Let Q(a,b,c)Q(a,b,c) be the image of the point P(3,2,1)P(3,2,1) in the line x−11=y2=z−11\frac{x- 1}{1}=\frac{y}{2}=\frac{z- 1}{1}. Then the distance of QQ from the line x−93=y−92=z−5−2\frac{x- 9}{3}=\frac{y- 9}{2}=\frac{z- 5}{- 2} is

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  17. Question 17

    4 Apr 2026 S2

    64
    If (2α+1,α2−3α,α−12)\left( 2\alpha+ 1,\alpha^{2}- 3\alpha,\frac{\alpha- 1}{2} \right) is the image of (α,2α,1)(\alpha,2\alpha,1) in the line x−23=y−12=z1\frac{x - 2}{3}=\frac{y - 1}{2}=\frac{z}{1}, then the possible value(s) of α\alpha is(are) :

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  18. Question 18

    5 Apr 2026 S2

    65
    If the distance of the point (a,2,5)(a,2,5) from the image of the point (1,2,7)(1,2,7) in the line x1=y−11=z−22\frac{x}{1}=\frac{y- 1}{1}=\frac{z- 2}{2} is 4, then the sum of all possible values of a is equal to :

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  19. Question 19

    6 Apr 2026 S1

    65
    Let the image of the point P(1,6,a)P(1,6,a) in the line L:x1=y−12=z−a+1 b, b>0L:\frac{x}{1}=\frac{y - 1}{2}=\frac{z - a + 1}{\text{ }b},\text{ }b > 0, be (a3,0,a+c)\left( \frac{a}{3},0,a + c \right). If S(α,β,γ),α>0S(\alpha,\beta,\gamma),\alpha> 0, is the point on L such that the distance of S from the foot of perpendicular from the point P on L is 2142\sqrt{14}, then α+β+γ\alpha+\beta+\gamma is equal to :

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