Question

The line $\dfrac{x+1}{2} = \dfrac{y-4}{4} = \dfrac{z-2}{5}$ passes through the point

• A. $(-3,\,0,\,-3)$
• B. $(3,\,0,\,5)$
• C. $(-3,\,0,\,5)$
• D. $(3,\,4,\,5)$
• E. $(3,\,-4,\,-5)$

Hint:

To check if a point lies on a 3D line in symmetric form, substitute the coordinates and verify all three ratios are equal. Setting each ratio to a parameter $\lambda$ gives the parametric equations of the line.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A point $(x_0,y_0,z_0)$ lies on the line $\dfrac{x-x_1}{l}=\dfrac{y-y_1}{m}=\dfrac{z-z_1}{n}$ if substituting the point makes all three ratios equal.

Step 2: Detailed Explanation:
Test $(-3,\,0,\,-3)$:
\[ \frac{-3+1}{2} = \frac{-2}{2} = -1, \quad \frac{0-4}{4} = \frac{-4}{4} = -1, \quad \frac{-3-2}{5} = \frac{-5}{5} = -1 \] All three ratios equal $-1$.

Step 3: Final Answer:
The line passes through $(-3,\,0,\,-3)$.