The lines x - 2 1 = y - 3 1 = 4 - z k and x - 1 k = y - 4 2 = z - 5 - 2 are mutually perpendicular if the value of k is:

Question

The lines $\frac{x - 2}{1} = \frac{y - 3}{1} = \frac{4 - z}{k}$ and $\frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{- 2}$ are mutually perpendicular if the value of $k$ is:

Smart Achievers · Accepted Answer

(1)

Ans. $- \frac{2}{3}$

Explanation:

First line is,

$\frac{x - 2}{1} = \frac{y - 3}{1} = \frac{z - 4}{- k}$

$∴$ Direction ratios of first line $= 1, 1, - k$

Second line is,

$\frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{- 2}$

$∴$ Direction ratios of second line $= k, 2, - 2$

We know, two lines are perpendicular to each other, if the dot product of their direction ratios is zero.

$i.e.,$ $1 \times k + 1 \times 2 + (- k) \times (- 2) = 0$

$\Rightarrow$ $k + 2 + 2 k = 0$

$\Rightarrow$ $3 k = - 2$

$\Rightarrow$ $k = - \frac{2}{3}$

Solution

Q.
The lines $\frac{x - 2}{1} = \frac{y - 3}{1} = \frac{4 - z}{k}$ and $\frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{- 2}$ are mutually perpendicular if the value of $k$ is:

1 $- \frac{2}{3}$

2 $\frac{2}{3}$

3 $- 2$

4 $2$

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Solution

Q. The lines x-21=y-31=4-zk and x-1k=y-42=z-5-2 are mutually perpendicular if the value of k is:

1 -23

2 23

3 -2