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

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

The direction ratios can be read only from the standard form \(\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\), where x, y and z each have coefficient 1. Factorising each part:

    \(5x - 25 = 5(x - 5)\)
    \(14 - 7y = -7(y - 2)\)
    \(35z = 35(z - 0)\)

So the given line is

    \(\displaystyle \frac{x - 5}{1/5} = \frac{y - 2}{-1/7} = \frac{z - 0}{1/35}\)

Its direction ratios are \(\frac{1}{5}, -\frac{1}{7}, \frac{1}{35}\). Multiplying by 35, they are 7, −5, 1.

A parallel line has the same direction ratios, so the line through (1, 2, −1) is

    \(\displaystyle \frac{x-1}{7} = \frac{y-2}{-5} = \frac{z+1}{1}\)

Why the other options are incorrect

• Option (1) \(\displaystyle\frac{x-1}{7}=\frac{y-2}{5}=\frac{z+1}{1}\) — Incorrect. Its direction ratios are 7, 5, 1. The y-ratio must be −5, because \(14 - 7y = -7(y - 2)\) has a negative coefficient of y.

• Option (3) \(x-1= y- 2 =z+1\) — Incorrect. Its direction ratios are 1, 1, 1, which are not proportional to 7, −5, 1.

• Option (4) \(\displaystyle\frac{x-1}{5}=\frac{y-2}{7}=\frac{z+1}{35}\) — Incorrect. It uses the coefficients 5, 7, 35 of x, y, z as direction ratios. The direction ratios are their reciprocals (with the sign of y), \(\frac{1}{5}, -\frac{1}{7}, \frac{1}{35}\).