If the angle between the lines, x2=y2=z1 and  5−x-2=7y−14P=z−34 is cos−1⁡23, then P is equal to

# If the angle between the lines, $\frac{x}{2}=\frac{y}{2}=\frac{z}{1}$ and  $\frac{5-x}{-2}=\frac{7y-14}{P}=\frac{z-3}{4}$ is ${\mathrm{cos}}^{-1}\left(\frac{2}{3}\right)$, then $P$ is equal to

1. A

-4/7

2. B

7/2

3. C

-7/4

4. D

2/7

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### Solution:

$\frac{x}{2}=\frac{y}{2}=\frac{z}{1},\frac{x-5}{2}=\frac{y-2}{p}{7}}=\frac{z-3}{4}$

$\mathrm{cos}\theta =\frac{2\left(2\right)+2\left(p}{7}\right)+1\left(4\right)}{\sqrt{4+4+1}\sqrt{4+\left({p}^{2}}{49}\right)+16}}=\frac{2}{3}⇒p=\frac{7}{2}$

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