Let PQR be a triangle such that P Q = - 2 i - j + 2 k and P R = a i + b j - 4 k , a , b . Let S be the point on QR which is equidistant from the lines PQ and PR. If | P R | = 9 a n d P S = i - 7 j + 2 k then the value of 3 a - 4 b is:

Question

Let PQR be a triangle such that $\vec{P Q} = - 2 \hat{i} - \hat{j} + 2 \hat{k}$ and $\vec{P R} = a \hat{i} + b \hat{j} - 4 \hat{k}, a, b \in ℤ$ . Let S be the point on QR which is equidistant from the lines PQ and PR. If $| \vec{P R} |$ $= 9 and \vec{P S} = \hat{i} - 7 \hat{j} + 2 \hat{k}$ then the value of $3 a - 4 b$ is: [2026]

Smart Achievers · Accepted Answer

(37)

$\vec{P S} = \hat{i} - 7 \hat{j} + 2 \hat{k}$

$\vec{P Q} = - 2 \hat{i} - \hat{j} + 2 \hat{k}$

$\vec{P R} = a \hat{i} + b \hat{j} - 4 \hat{k}$

$\vec{P S} = λ \vec{P R} + \vec{P Q}$

$\hat{i} - 7 \hat{j} + 2 \hat{k} = λ (\frac{a \hat{i} + b \hat{j} - 4 \hat{k}}{9} + \frac{- 2 \hat{i} - \hat{j} + 2 \hat{k}}{3})$

$\hat{i} - 7 \hat{j} + 2 \hat{k} = \frac{λ}{9} (a \hat{i} + b \hat{j} - 4 \hat{k}) - 2 \hat{i} - \hat{j} + 2 \hat{k}$

$\hat{i} - 7 \hat{j} + 2 \hat{k} = \frac{λ}{9} (a \hat{i} + 6 \hat{j} - 4 \hat{k} - 6 \hat{i} - 3 \hat{j} + 6 \hat{k})$

$\hat{i} - 7 \hat{j} + 2 \hat{k} = \frac{λ}{9} (a - 6) \hat{i} + \frac{λ}{9} (b - 3) \hat{j} + \frac{2 λ}{9} \hat{k}$

$\frac{2 λ}{9} = 2$

$λ = 9, a - 6 = 1$

$a = 7$

$b - 3 = - 7$

$b = - 4$

$3 a - 4 b = (21 + 16) = 37$

Solution

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Solution

Q. Let PQR be a triangle such that PQ→=-2i^-j^+2k^ and PR→=ai^+bj^-4k^, a,b∈ℤ. Let S be the point on QR which is equidistant from the lines PQ and PR. If |PR→|=9 and PS→=i^-7j^+2k^ then the value of 3a-4b is: [2026]

Ans. (37) PS→=i^-7j^+2k^ PQ→=-2i^-j^+2k^ PR→=ai^+bj^-4k^ PS→=λPR→+PQ→ i^-7j^+2k^ =λ(ai^+bj^-4k^9+-2i^-j^+2k^3) i^-7j^+2k^ =λ9(ai^+bj^-4k^) -2i^-j^+2k^ i^-7j^+2k^ =λ9(ai^+6j^-4k^-6i^-3j^+6k^) i^-7j^+2k^ =λ9(a-6)i^+λ9(b-3)j^+2λ9k^ 2λ9=2 λ=9, a-6=1 a=7 b-3=-7 b=-4 3a-4b=(21+16)=37

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