Question

If the line \[ \frac{x - 3}{a} = \frac{y - 4}{b} = \frac{z - 5}{c} \] is parallel to the line \[ \frac{x}{5} = \frac{y}{3} = \frac{z}{2} \] then:

• A. \( 5a + 3b + 2c = 0 \)
• B. \( \frac{5}{a} = \frac{3}{b} = \frac{2}{c} \)
• C. \( 5a = 3b = 2c \)
• D. none of these

Hint:

Two lines are parallel if their direction ratios are proportional, i.e., \[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

Answer 1

The Correct Option is C

For two lines to be parallel, their direction ratios must be proportional. The direction ratios of the first line are \( (a, b, c) \). The direction ratios of the second line are \( (5, 3, 2) \). Therefore, \[ \frac{a}{5} = \frac{b}{3} = \frac{c}{2} \] which implies \[ 5a = 3b = 2c \] Thus, the correct option is: \[ \boxed{5a = 3b = 2c} \]