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If the lines (x-1)/(-3)=(y-2)/(2k)=(z-3...

If the lines `(x-1)/(-3)=(y-2)/(2k)=(z-3)/2`and `(x-1)/(3k)=(y-1)/1=(z-6)/(-5)`are perpendicular, find the value of k.

A

`-11/19`

B

`-8/7`

C

`-10/7`

D

`-9/7`

Text Solution

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The correct Answer is:
To find the value of \( k \) such that the lines \[ \frac{x-1}{-3} = \frac{y-2}{2k} = \frac{z-3}{2} \] and \[ \frac{x-1}{3k} = \frac{y-1}{1} = \frac{z-6}{-5} \] are perpendicular, we can follow these steps: ### Step 1: Identify the direction ratios of both lines. For the first line, the direction ratios can be extracted from the equation: - The direction ratios are \( (-3, 2k, 2) \). For the second line, we similarly extract the direction ratios: - The direction ratios are \( (3k, 1, -5) \). ### Step 2: Set up the dot product condition for perpendicularity. Two lines are perpendicular if the dot product of their direction ratios is zero. Therefore, we set up the equation: \[ (-3)(3k) + (2k)(1) + (2)(-5) = 0 \] ### Step 3: Simplify the dot product equation. Calculating the dot product gives: \[ -9k + 2k - 10 = 0 \] Combine like terms: \[ -7k - 10 = 0 \] ### Step 4: Solve for \( k \). Rearranging the equation gives: \[ -7k = 10 \] Dividing both sides by -7: \[ k = -\frac{10}{7} \] ### Final Answer: The value of \( k \) is \[ \boxed{-\frac{10}{7}}. \] ---
To find the value of \( k \) such that the lines \[ \frac{x-1}{-3} = \frac{y-2}{2k} = \frac{z-3}{2} \] and ...
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