Question

Let the image of the point P(1, 2, 3) in the line
$L:\frac{x−6}{3}=\frac{y−1}{2}=\frac{z−2}{3}$
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 ________.

Answer 1

The correct answer is 125
The point dividing PQ in the ratio 1 : 3 will be mid-point of P & foot of perpendicular from P on the line.
∴ Let a point on line be λ
$⇒\frac{x−6}{3}=\frac{y−1}{2}=\frac{z−2}{3}=λ$
$⇒P^′(3λ+6,2λ+1,3λ+2)$
as P′ is foot of perpendicular
(3λ + 5)3 + (2λ – 1)2 + (3λ – 1)3 = 0
⇒ 22λ + 15 – 2 – 3 = 0
⇒λ=−5/11
$∴P^′(\frac{51}{11},\frac{1}{11},\frac{7}{11})$
Mid-point of PP’
$≡(\frac{\frac{51}{11}+1}{2},\frac{\frac{1}{11}+2}{2},\frac{\frac{7}{11}+3}{2})≡(\frac{62}{22},\frac{23}{22},\frac{40}{22})≡(α,β,γ)$
⇒ 22(α + β + γ)
= 62 + 23 + 40 = 125