Discussion
The equation of the straight line joining the points (1, 2,3) and (3, 4, k) is \(\frac{x-3}{1} = \frac{y-4}{1} = \frac{z-k}{5}\) . Then the value of \(k\) is
Answer: E
The value of k is 13.
The line through (1, 2, 3) and (3, 4, k) has direction ratios \((3 - 1, 4 - 2, k - 3) = (2, 2, k - 3)\). These must be proportional to the given denominators (1, 1, 5):
\[ \begin{aligned} \frac{2}{1} = \frac{2}{1} &= \frac{k - 3}{5} \\ k - 3 &= 10 \Rightarrow k = 13 \end{aligned} \]
The given equation already passes through (3, 4, k), so only the direction matters. 10 is the value of k - 3, the step before the answer, and the smaller values do not make the direction ratios proportional.
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