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: Find direction ratios.
From $A(3,4,-7)$ to $B(1,-1,6)$ the direction is $(1-3,\ -1-4,\ 6+7) = (-2,-5,13)$.

Step 2: Use point A.
With base point $A(3,4,-7)$ and the parameter $\lambda$, write each coordinate.

Step 3: Write the parametric form.
$x = 3 - 2\lambda$, $y = 4 - 5\lambda$, $z = -7 + 13\lambda$.
\[ \boxed{\text{Option 4}} \]