Question

The lines $\frac{6x-6}{18}=\frac{y+1}{3}=\frac{z-1}{5}$ and $\frac{3x+6}{12}=\frac{y-1}{3}=\frac{z+1}{2}$ are...

• A. intersecting at point (1, -1, 2)
• B. intersecting at right angles
• C. do not intersect
• D. intersecting at point (3, 1, -1)

Hint:

Always normalize the coefficients of $x, y, z$ to 1 before finding direction ratios.

Answer 1

The Correct Option is C

Step 1: Concept
Reduce the lines to standard form $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$.

Step 2: Analysis
Line 1: $\frac{x-1}{3} = \frac{y+1}{3} = \frac{z-1}{5}$ (Direction: $3, 3, 5$).
Line 2: $\frac{x+2}{4} = \frac{y-1}{3} = \frac{z+1}{2}$ (Direction: $4, 3, 2$).

Step 3: Calculation
To check for intersection, equate the general points:
$(3\lambda+1, 3\lambda-1, 5\lambda+1) = (4\mu-2, 3\mu+1, 2\mu-1)$.
From y-coordinates: $3\lambda - 1 = 3\mu + 1 \implies 3\lambda - 3\mu = 2$.
From x-coordinates: $3\lambda + 1 = 4\mu - 2 \implies 3\lambda - 4\mu = -3$.
Subtracting gives $\mu = 5$ and $\lambda = 17/3$.
Check z: $5(17/3)+1 = 88/3$ vs $2(5)-1 = 9$. Not equal.

Step 4: Conclusion
Since the coordinates do not satisfy all three equations, the lines do not intersect. Final Answer: (C)