Question

The parametric equations of a line passing through the points $A(3,4,-7)$ and $B(1,-1,6)$ are

• A. $x = 3 + \lambda, y = -1 + 4\lambda, z = -7 + 6\lambda$
• B. $x = -2 + 3\lambda, y = -5 + 4\lambda, z = 13 - 7\lambda$
• C. $x = 1 + 3\lambda, y = -1 + 4\lambda, z = 6 - 7\lambda$
• D. $x = 3 - 2\lambda, y = 4 - 5\lambda, z = -7 + 13\lambda$

Hint:

An easy multiple-choice trick is to check the coefficients of $\lambda$, which must be proportional to the direction ratios. The coordinate differences are $-2$, $-5$, and $13$. Only option (D) features those exact multipliers $(-2\lambda, -5\lambda, 13\lambda)$, which isolates the correct answer immediately without extra algebra!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given two points in three-dimensional space, $A(3, 4, -7)$ and $B(1, -1, 6)$. We need to determine the parametric equations representing the straight line passing through both of them.

Step 2: Key Formula or Approach:
The symmetric and parametric vector equation of a line passing through a fixed point $(x_1, y_1, z_1)$ with direction ratios $(l, m, n)$ is given by: $$ \frac{x - x_1}{l} = \frac{y - y_1}{m} = \frac{z - z_1}{n} = \lambda $$ The direction ratios $(l, m, n)$ for a line passing through two points $A$ and $B$ can be found by taking the coordinate differences: $$ l = x_2 - x_1, \quad m = y_2 - y_1, \quad n = z_2 - z_1 $$

Step 3: Detailed Explanation:
Let's first compute the direction ratios of the line using the points $A(3, 4, -7)$ and $B(1, -1, 6)$: $$ l = 1 - 3 = -2 $$ $$ m = -1 - 4 = -5 $$ $$ n = 6 - (-7) = 6 + 7 = 13 $$ Thus, the direction ratios are $(-2, -5, 13)$. Now, using the coordinates of the first point $A(3, 4, -7)$ as our reference base, we set up the parametric ratios equal to the parameter $\lambda$: $$ \frac{x - 3}{-2} = \frac{y - 4}{-5} = \frac{z - (-7)}{13} = \lambda $$ Isolating each coordinate variable independently in terms of $\lambda$:

• $\frac{x - 3}{-2} = \lambda \implies x - 3 = -2\lambda \implies x = 3 - 2\lambda$

• $\frac{y - 4}{-5} = \lambda \implies y - 4 = -5\lambda \implies y = 4 - 5\lambda$

• $\frac{z + 7}{13} = \lambda \implies z + 7 = 13\lambda \implies z = -7 + 13\lambda$
This matches the set of parametric equations in option (D).

Step 4: Final Answer:
The parametric equations of the line are perfectly represented by option (D).