Let P(alpha, 4, 7) and Q(3, beta, 8) be two p | vedclass

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(A) Given points are $P(\alpha, 4, 7)$ and $Q(3, \beta, 8)$.Since the $YZ$-plane divides the line segment $PQ$ in the ratio $2:3$,the $x$-coordinate o

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$\sqrt{107}$

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(A) Given points are $P(\alpha, 4, 7)$ and $Q(3, \beta, 8)$.Since the $YZ$-plane divides the line segment $PQ$ in the ratio $2:3$,the $x$-coordinate of the point of division is zero.Using the section formula: $\frac{2(3) + 3(\alpha)}{2+3} = 0 \Rightarrow 6 + 3\alpha = 0 \Rightarrow \alpha = -2$.Since the $ZX$-plane divides the line segment $PQ$ in the ratio $4:5$,the $y$-coordinate of the point of division is zero.Using the section formula: $\frac{4(\beta) + 5(4)}{4+5} = 0 \Rightarrow 4\beta + 20 = 0 \Rightarrow \beta = -5$.Thus,the points are $P(-2, 4, 7)$ and $Q(3, -5, 8)$.The length of the line segment $PQ$ is given by the distance formula:$PQ = \sqrt{(3 - (-2))^2 + (-5 - 4)^2 + (8 - 7)^2}$$PQ = \sqrt{(5)^2 + (-9)^2 + (1)^2}$$PQ = \sqrt{25 + 81 + 1} = \sqrt{107}$.

Let $P(\alpha, 4, 7)$ and $Q(3, \beta, 8)$ be two points. If the $YZ$-plane divides the line segment joining $P$ and $Q$ in the ratio $2:3$ and the $ZX$-plane divides the line segment joining $P$ and $Q$ in the ratio $4:5$,then the length of the line segment $PQ$ is:

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