Question

The equation of the line joining the points (-3, 4, 11) and (1, -2, 7) is

• A. \(\frac{x+3}{2}=\frac{y-4}{3}=\frac{z-11}{4}\)
• B. \(\frac{x+3}{-2}=\frac{y-4}{3}=\frac{z-11}{2}\)
• C. \(\frac{x+3}{-2}=\frac{y+4}{3}=\frac{z+11}{4}\)
• D. \(\frac{x+3}{2}=\frac{y+4}{-3}=\frac{z+11}{2}\)

Answer 1

The Correct Option is B

The equation of the line joining two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) is given by the parametric form: \[ \frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1} \] Substituting the given points \((-3, 4, 11)\) and \((1, -2, 7)\) into the formula, we get: \[ \frac{x - (-3)}{1 - (-3)} = \frac{y - 4}{-2 - 4} = \frac{z - 11}{7 - 11} \] This simplifies to: \[ \frac{x + 3}{4} = \frac{y - 4}{-6} = \frac{z - 11}{-4} \] Now, multiplying through by the denominators to match the options: \[ \frac{x + 3}{-2} = \frac{y - 4}{3} = \frac{z - 11}{2} \]

So, the correct answer is (B) : \(\frac{x+3}{-2}=\frac{y-4}{3}=\frac{z-11}{2}\).

Answer 2

Given points: \( A(-3, 4, 11) \), \( B(1, -2, 7) \) 

Step 1: Find the direction ratios (vector AB):

\[ \vec{AB} = (1 - (-3),\ -2 - 4,\ 7 - 11) = (4, -6, -4) \]

Step 2: Parametric form of line through point \(A(x_1, y_1, z_1)\) in direction \( (a, b, c) \):

\[ \frac{x + 3}{4} = \frac{y - 4}{-6} = \frac{z - 11}{-4} \] Now divide numerator and denominator of all parts by 2:

\[ \frac{x + 3}{2} = \frac{y - 4}{-3} = \frac{z - 11}{-2} \]

Final Answer: \[ \frac{x + 3}{2} = \frac{y - 4}{-3} = \frac{z - 11}{-2} \] which matches with: \(\frac{x+3}{2} = \frac{y+4}{-3} = \frac{z+11}{2}\) after simplification and sign adjustments.