First slide

Straight lines in 3D

Question

The distance between the points $(5, 1, 7)$ and $(c, - 5, 1)$ is $9$ then $c =$

Easy

A

2
B

$- 2$
C

3
D

$- 4$

Solution

The distance between two points $A (x_{1}, y_{1}, z_{1})$ and $B (x_{2}, y_{2}, z_{2})$ is $d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} + {(z_{2} - z_{1})}^{2}}$
Hence the distance between the points $(5, 1, 7)$ and $(c, - 5, 1)$ is $d = \sqrt{{(c - 5)}^{2} + 36 + 36}$
It implies that
$\begin{matrix} 9 = \sqrt{{(c - 5)}^{2} + 36 + 36} \\ 81 = {(c - 5)}^{2} + 72 \\ {(c - 5)}^{2} = 9 \\ c = 5 \pm 3 \\ = 8, 2 \end{matrix}$
Therefore, $c = 8, 2$

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