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 P Q in the ratio 1 : 3 . Then the value of 22 α + β + γ is equal to

Question

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 &#945; , &#946; , &#947; be a point that divides internally the line segment P Q in the ratio 1 : 3 . Then the value of 22 &#945; + &#946; + &#947; is equal to

Aicharya · Accepted Answer

Plotting the rough graph of the given data we have,

Let $M$ be the mid-point of $P Q$

$∴ M = (3 λ + 6, 2 λ + 1, 3 λ + 2)$

Now, $\vec{P M} = (3 λ + 5) \hat{i} + (2 λ - 1) \hat{j} + (3 λ - 1) \hat{k}$

$∵ \vec{P M} ⊥ (3 \hat{i} + 2 \hat{j} + 3 \hat{k})$

$∴ 3 (3 λ + 5) + 2 (2 λ - 1) + 3 (3 λ - 1) = 0$

$λ = \frac{- 5}{11}$

$∴ M (\frac{51}{11}, \frac{1}{11}, \frac{7}{11})$

Since $R$ is mid-point of $P M$

So by using midpoint formula in $P & M$ we get,

$R = (\frac{62}{22}, \frac{23}{22}, \frac{40}{22})$ , now comparing with $R (α, β, γ)$

We get, $α = \frac{62}{22}, β = \frac{23}{22} & γ = \frac{40}{22}$

So, $22 (α + β + γ) = 125$

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 ,…

Solution

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 &#945; ,…

Solution

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 ,…