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Plane through the line of intersection of two planes (P1 + k P2 = 0) with one more condition

This type of question, asked 15 times: 2021 Paper 2, 2021 Paper 7, 2021 Paper 18, 2021 Paper 19, 2021 Paper 20, 2021 Paper 21, 2021 Paper 22, 28 Jun 2022, 26 Jul 2022, 24 Jan 2023, 25 Jan 2023, 1 Feb 2023, 6 Apr 2023, 8 Apr 2023, 13 Apr 2023. All 16 questions below, oldest first.

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  1. Question 1

    2021 Paper 2

    65
    The vector equation of the plane passing through the intersection of the planes r→⋅(i^+j^+k^)=1\overrightarrow{r}\cdot (\widehat{i}+\widehat{j}+\widehat{k}) = 1 and r→⋅(i^−2j^)=−2\overrightarrow{r}\cdot (\widehat{i}- 2\widehat{j}) = - 2, and the point (1,0,2)(1,0,2) is :

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  2. Question 2

    2021 Paper 7

    89
    If the equation of the plane passing through the line of intersection of the planes 2x−7y+4z−3=0,3x−5y+4z+11=02x - 7y + 4z - 3 = 0,3x - 5y + 4z + 11 = 0 and the point (−2,1,3)( - 2,1,3) is ax + by +cz−7=0+ cz - 7 = 0, then the value of 2a+b+c−72a + b + c - 7 is

  3. Question 3

    2021 Paper 18

    75
    A plane PP contains the line x+2y+3z+1=0=x−y−z−6x + 2y + 3z + 1 = 0 = x - y - z - 6, and is perpendicular to the plane −2x+y+z+8=0- 2x+y+z+ 8 = 0. Then which of the following points lies on PP ?

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  4. Question 4

    2021 Paper 19

    76
    Let PP be the plane passing through the point (1,2,3)(1,2,3) and the line of intersection of the planes r→⋅(i^+j^+4k^)=16\overrightarrow{r}\cdot (\widehat{i}+\widehat{j}+ 4\widehat{k}) = 16 and r→⋅(−i^+j^+k^)=6\overrightarrow{r}\cdot ( -\widehat{i}+\widehat{j}+\widehat{k}) = 6. Then which of the following points does NOT lie on P?

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  5. Question 5

    2021 Paper 20

    69
    Equation of a plane at a distance 221\sqrt{\frac{2}{21}} from the origin, which contains the line of intersection of the planes x−y−z−1=0x - y - z - 1 = 0 and 2x+y−3z+4=02x + y - 3z + 4 = 0, is:

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  6. Question 6

    2021 Paper 21

    67
    The equation of the plane through the line of intersection of the planes r→⋅(i^+j^+k^)=1\overrightarrow{r}\cdot (\widehat{i}+\widehat{j}+\widehat{k}) = 1 and r→⋅(2i^+3j^−k^)+4=0\overrightarrow{r}\cdot (2\widehat{i}+ 3\widehat{j}-\widehat{k}) + 4 = 0 and parallel to the xx-axis is:

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  7. Question 7

    2021 Paper 22

    64
    Let the equation of the plane, that passes through the point (1,4,−3)(1,4,-3) and contains the line of intersection of the planes 3x−2y+4z−7=03x- 2y+ 4z- 7 = 0 and x+5y−2z+9=0x + 5y - 2z + 9 = 0, be αx+βy+γz+3=0\alpha x +\beta y +\gamma z + 3 = 0, then α+β+γ\alpha+\beta+\gamma is equal to:

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  8. Question 8

    28 Jun 2022

    77
    Let the plane P:r→⋅a→=dP:\overrightarrow{r}\cdot\overrightarrow{a}= d contain the line of intersection of the planes r→⋅(i^+3j^−k^)=6\overrightarrow{r}\cdot (\widehat{i}+ 3\widehat{j}-\widehat{k}) = 6 and r→⋅(−6i^+5j^−k^)=7\overrightarrow{r}\cdot ( - 6\widehat{i}+ 5\widehat{j}-\widehat{k}) = 7. If the plane PP passes through the point (2,3,12)\left( 2,3,\frac{1}{2} \right), then the value of ∣13a→∣2d2\frac{|13\overrightarrow{a}|^{2}}{d^{2}} is equal to

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  9. Question 9

    28 Jun 2022

    76
    The acute angle between the planes P1P_{1} and P2P_{2}, when P1P_{1} and P2P_{2} are the planes passing through the intersection of the planes 5x+8y+13z−29=05x + 8y + 13z - 29 = 0 and 8x−7y+z−20=08x - 7y + z - 20 = 0 and the points (2,1,3)(2,1,3) and (0,1,2)(0,1,2), respectively, is

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  10. Question 10

    26 Jul 2022

    177
    The plane passing through the line L:lx−y+3(1−l)L:\mathcal{l}x - y + 3(1 -\mathcal{l)} z=1,x+2y−z=2z = 1,x + 2y - z = 2 and perpendicular to the plane 3x+2y+z=63x + 2y + z = 6 is 3x−8y+7z=43x - 8y + 7z = 4. If θ\theta is the acute angle between the line LL and the yy-axis, then 415 cos⁡2θ\cos^{2}\theta is equal to

  11. Question 11

    24 Jan 2023

    162
    Let the plane containing the line of intersection of the planes P1:x+(λ+4)y+z=1P1:x + (\lambda + 4)y + z = 1 and P2:2x+y+P2:2x + y + z=2z = 2 pass through the points (0,1,0)(0,1,0) and (1,0,1)(1,0,1). Then the distance of the point (2λ,λ,−λ)(2\lambda,\lambda, - \lambda) from the plane P2P2 is

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  12. Question 12

    25 Jan 2023

    84
    Let the equation of the plane passing through the line x−2y−z−5=0=x+y+3z−5x - 2y - z - 5 = 0 = x + y + 3z - 5 and parallel to the line x+y+2z−7=0=2x+3y+z−2x + y + 2z - 7 = 0 = 2x + 3y + z - 2 be ax+by+cz=65ax + by + cz = 65. Then the distance of the point (a,b,c)(a,b,c) from the plane 2x+2y−z+16=02x + 2y - z + 16 = 0 is

  13. Question 13

    1 Feb 2023

    161
    Let the plane PP pass through the intersection of the planes 2x+3y−z=22x + 3y - z = 2 and x+2y+3z=6x + 2y + 3z = 6 and be perpendicular to the plane 2x+y−z=02x + y - z = 0. If dd is the distance of PP from the point (−7,1,1)(-7,1,1), then d2d^{2} is equal to:

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  14. Question 14

    6 Apr 2023

    163
    A plane PP contains the line of intersection of the plane r→⋅(i^+j^+k^)=6\overrightarrow{r}\cdot (\widehat{i}+\widehat{j}+\widehat{k}) = 6 and r→⋅(2i^+3j^+4k^)=−5\overrightarrow{r}\cdot (2\widehat{i}+ 3\widehat{j}+ 4\widehat{k}) = - 5. If PP passes through the point (0,2,−2)(0,2, - 2), then the square of distance of the point (12,12,18)(12,12,18) from the plane PP is

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  15. Question 15

    8 Apr 2023

    62
    If the equation of the plane containing the line x+2y+3z−4=0=2x+y−z+5x + 2y + 3z - 4 = 0 = 2x + y - z + 5 and perpendicular to the plane r→=(i^−j^)+λ(i^+j^+k^)+μ(i^−2j^+3k)\overrightarrow{r}= (\widehat{i}-\widehat{j}) + \lambda(\widehat{i}+\widehat{j}+\widehat{k}) + \mu(\widehat{i}- 2\widehat{j}+ 3k) is ax+by+cz=4ax + by + cz = 4, then (a−b+c)(a - b + c) is equal to

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  16. Question 16

    13 Apr 2023

    80
    Let the equation of plane passing through the line of intersection of the planes x+2y+az=2x + 2y + az = 2 and x−y+z=3x - y + z = 3 be 5x−11y+bz=6a−15x - 11y + bz = 6a - 1. For c∈Zc \in Z, if the distance of this plane from the point (a,−c,c)(a, - c,c) is 2a\frac{2}{\sqrt{a}}, then a+bc\frac{a + b}{c} is equal to

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