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(B) Let the line be $L: \frac{x - 1}{2} = \frac{y}{9} = \frac{z}{5} = \lambda$. Any point $Q$ on the line is given by $(2\lambda + 1, 9\lambda, 5\lamb

Vedclass · Accepted Answer

$\sqrt{\frac{2109}{110}}$

Vedclass · Answer

(B) Let the line be $L: \frac{x - 1}{2} = \frac{y}{9} = \frac{z}{5} = \lambda$. Any point $Q$ on the line is given by $(2\lambda + 1, 9\lambda, 5\lambda)$. The direction ratios of the line $PQ$ are $(2\lambda + 1 - 5, 9\lambda - 4, 5\lambda + 1) = (2\lambda - 4, 9\lambda - 4, 5\lambda + 1)$. Since $PQ$ is perpendicular to the line $L$ with direction ratios $(2, 9, 5)$,their dot product is zero: $2(2\lambda - 4) + 9(9\lambda - 4) + 5(5\lambda + 1) = 0$. $4\lambda - 8 + 81\lambda - 36 + 25\lambda + 5 = 0$. $110\lambda - 39 = 0 \implies \lambda = \frac{39}{110}$. The coordinates of $Q$ are $(2(\frac{39}{110}) + 1, 9(\frac{39}{110}), 5(\frac{39}{110})) = (\frac{188}{110}, \frac{351}{110}, \frac{195}{110})$. The length $PQ = \sqrt{(2\lambda - 4)^2 + (9\lambda - 4)^2 + (5\lambda + 1)^2}$. Substituting $\lambda = \frac{39}{110}$,we get $PQ = \sqrt{\frac{2109}{110}}$.

The length of the perpendicular drawn from the point $P(5, 4, -1)$ to the line $\frac{x - 1}{2} = \frac{y}{9} = \frac{z}{5}$ is:

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