Let the image of the point P(1, 2, 3) in the line L:(x - 6)/3 = (y - 1)/2 = (z - 2)/3 be Q. Let R (α, β, γ) be a point that divides internally the line segment PQ in the ratio 1 : 3. Then the value of 22 (α + β + γ) is equal to __________.

Solution

Maths sir · 4 steps

Find the foot of the perpendicular first. Then R turns out to be just the midpoint of P and that foot.

1. Foot of perpendicular. A general point on L is M = (6+3t, 1+2t, 2+3t). For the foot, PM⃗ = (5+3t, -1+2t, -1+3t) must be perpendicular to the direction (3, 2, 3).

   3(5+3t) + 2(-1+2t) + 3(-1+3t) = 10 + 22t = 0 ⇒ t = -5/11

2. Locate M and Q. So M = (51/11, 1/11, 7/11). M is the midpoint of PQ, so Q = 2M - P = (91/11, -20/11, -19/11).
3. Section formula. R divides PQ internally in 1 : 3 from P, so R = (3P + Q)/4. Since Q = 2M - P, this is (2P + 2M)/4 = (P + M)/2, the midpoint of P and M.
4. So. The coordinates of P add to 6 and those of M add to 59/11, so α + β + γ = 1/2(6 + 59/11) = 125/22. Then 22(α + β + γ) = 125.

The common slip: 111. Putting the weights the wrong way round, R = (P + 3Q)/4, gives 111 instead of 125.

The idea to keep. For the image of a point in a line, find the foot M from (M − P) · direction = 0, then use Q = 2M − P. In the section formula, the weight on P is the part of the ratio next to Q.

Answer from the NTA final answer key. Solution by PrepWiser.