Question

If the lines $\frac{1-x}{3}=\frac{7y-14}{2\lambda}=\frac{z-3}{2}$ and $\frac{7-7x}{3\lambda}=\frac{y-5}{1}=\frac{6-z}{5}$ are at right angles, then $\lambda =$

• A. $-\frac{70}{11}$
• B. $\frac{70}{11}$
• C. $\frac{11}{70}$
• D. $-\frac{11}{70}$

Hint:

The most common mistake in 3D geometry problems is immediately extracting direction ratios from non-standard line equations! Always guarantee the coefficients of $x$, $y$, and $z$ in the numerator are exactly $+1$ before pulling your denominator values.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given the Cartesian equations of two lines in 3D space. The lines intersect at right angles (they are completely perpendicular). We must find the value of the unknown variable $\lambda$.

Step 2: Key Formula or Approach:
First, the equations of both lines must be converted into standard symmetric form: $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$, where the coefficients of $x$, $y$, and $z$ in the numerators must be exactly $+1$.
Once in this standard form, the denominators represent the valid direction ratios $\langle a, b, c \rangle$ of the lines.
If two lines are perpendicular, the dot product of their respective direction vectors must be zero: $a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$.

Step 3: Detailed Explanation:
Let's rewrite Line 1 into the standard symmetric form:
$$\frac{-(x-1)}{3} = \frac{7(y-2)}{2\lambda} = \frac{z-3}{2}$$ $$\frac{x-1}{-3} = \frac{y-2}{\frac{2\lambda}{7}} = \frac{z-3}{2}$$ The direction ratios of Line 1 are: $\vec{d}_1 = \langle -3, \frac{2\lambda}{7}, 2 \rangle$.
Now, let's rewrite Line 2 into the standard symmetric form:
$$\frac{-7(x-1)}{3\lambda} = \frac{y-5}{1} = \frac{-(z-6)}{5}$$ $$\frac{x-1}{\frac{-3\lambda}{7}} = \frac{y-5}{1} = \frac{z-6}{-5}$$ The direction ratios of Line 2 are: $\vec{d}_2 = \langle \frac{-3\lambda}{7}, 1, -5 \rangle$.
Since the lines are perpendicular, set the dot product of their direction vectors to zero:
$$(-3)\left(\frac{-3\lambda}{7}\right) + \left(\frac{2\lambda}{7}\right)(1) + (2)(-5) = 0$$ $$\frac{9\lambda}{7} + \frac{2\lambda}{7} - 10 = 0$$ $$\frac{11\lambda}{7} = 10$$ $$11\lambda = 70 \implies \lambda = \frac{70}{11}$$

Step 4: Final Answer:
The value of $\lambda$ is $\frac{70}{11}$, corresponding to option (B).