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Ratio in which a plane divides the segment joining two points

This type of question, asked 3 times: 2021 Paper 5, 29 Jan 2023, 1 Feb 2023. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 5

    73
    Let PP be a plane lx+my+nz=0lx + my + nz = 0 containing the line, 1−x1=y+42=z+23\frac{1 -x}{1}=\frac{y+ 4}{2}=\frac{z+ 2}{3}. If plane PP divides the line segment ABAB joining points A(−3,−6,1)A( - 3, - 6,1) and B(2,4,−3)B(2,4, - 3) in ratio k:1k:1 then the value of kk is equal to:

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  2. Question 2

    29 Jan 2023

    158
    The plane 2x−y+z=42x - y + z = 4 intersects the line segment joining the points A(a,−2,4)A(a, - 2,4) and B(2,b,−3)B(2,b, - 3) at the point CC in the ratio 2: 1 and the distance of the point CC from the origin is 5\sqrt{5}. If ab<0ab < 0 and PP is the point (a−b,b,2 b−a)(a - b,b,2\text{ }b - a) then CP2CP^{2} is equal to

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  3. Question 3

    1 Feb 2023

    175
    The point of intersection CC of the plane 8x+y+2z=08x+y+2z=0 and the line joining the points A(−3,−6,1)A(-3,-6,1) and B(2,−4,−3)B(2,-4,-3) divides the line segment ABAB internally in the ratio 14:314:3. If a,b,ca,b,c (where ∣a∣,∣b∣,∣c∣|a|,|b|,|c| are coprime) are the direction ratios of the perpendicular from the point CC on the line 1−x1=y+42=z+23\frac{1-x}{1}=\frac{y+4}{2}=\frac{z+2}{3}, then ∣a+b+c∣|a+b+c| is equal to