The Cartesian equation of the line passing through the point (1, 2, -…

The correct answer is Option (2) → $\frac{x-1}{7}=\frac{y-2}{-5}=\frac{z+1}{1}$

Given line: $\frac{5x - 25}{1} = \frac{14 - 7y}{1} = 35z$

Rewrite to extract direction ratios:

First, express in standard form: $5x - 25 = 14 - 7y = 35z = k$

From first part: $5x - 25 = k \Rightarrow x = \frac{k + 25}{5}$

From second part: $14 - 7y = k \Rightarrow y = \frac{14 - k}{7}$

From third part: $35z = k \Rightarrow z = \frac{k}{35}$

Hence direction ratios are proportional to $\left(\frac{1}{5}, -\frac{1}{7}, \frac{1}{35}\right)$ or equivalently $(7, -5, 1)$ after multiplying through by 35.

Required line passes through (1, 2, -1) and is parallel to $(7, -5, 1)$.

∴ Cartesian equation of the line is:

$\frac{x - 1}{7} = \frac{y - 2}{-5} = \frac{z + 1}{1}$

Final Answer:

$\frac{x - 1}{7} = \frac{y - 2}{-5} = \frac{z + 1}{1}$