Question

Find the Cartesian equation of the line passing through (−1, −1, 2) and parallel to the line 2x − 2 = 3y + 1 = 6z – 2

Answer 1

2x − 2 = 3y + 1 = 6z – 2     .......[Given]

∴ 2(x − 1) = `3(y + 1/3)`

= `6(z - 2/3)`

∴ `(x - 1)/(1/2) = (y - (-1/3))/(1/3)`

= `(z - 1/3)/(1/6)` 

Direction ratios of given line are `1/2, 1/3, 1/6`.

Since the required line is parallel to the given line, direction ratios of the required line will be `1/2, 1/3, 1/6`.

Equation of a line passing through the point (x₁, y₁, z₁) and having direction ratios (a, b, c) is

`(x - x_1)/"a" = (y - y_1)/"b" = (z - z_1)/"c"`

∴ `(x -(-1))/(1/2) = (y - (-1))/(1/3) = (z - 2)/(1/6)`

∴ `(x + 1)/(1/2 xx 6) = (y + 1)/(1/3 xx 6) = (z - 2)/(1/6 xx 6)`

∴ `(x + 1)/3 = (y + 1)/2 = (z - 2)/1`,

which is the required cartesian equation of the line.