which of the following pairs of points have a distance  43 ?

# which of the following pairs of points have a distance  $\sqrt{43}$ ?

1. A

(2,3, 5) and (4,3,1)

2. B

(-3,7,2) and (2, 4, -1)

3. C

(-1, 3, -4) and (1, -3, 4)

4. D

(2, -1, 3) and (-2,1, 3)

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### Solution:

(a) Let the given points are A(2,3,5) and B(4,3,1).

Required distance

(b) Let the given points are A(- 3,7,2) and B(2,4, -1).

Here,

$\begin{array}{r}{\mathrm{x}}_{1}=-3,{\mathrm{y}}_{1}=7,{\mathrm{z}}_{1}=2\\ {\mathrm{x}}_{2}=2,{\mathrm{y}}_{2}=4,{\mathrm{z}}_{2}=-1\end{array}$

Required distance

$\begin{array}{c}\mathrm{AB}=\sqrt{{\left({\mathrm{x}}_{2}-{\mathrm{x}}_{1}\right)}^{2}+{\left({\mathrm{y}}_{2}-{\mathrm{y}}_{1}\right)}^{2}+{\left({\mathrm{z}}_{2}-{\mathrm{z}}_{1}\right)}^{2}}\\ =\sqrt{\left[2-\left(-3\right){\right]}^{2}+\left(4-7{\right)}^{2}+\left(-1-2{\right)}^{2}}\\ =\sqrt{\left(2+3{\right)}^{2}+\left(4-7{\right)}^{2}+\left(-1-2{\right)}^{2}}\\ =\sqrt{25+9+9}=\sqrt{43}\end{array}$

(c)  Let the given ,points  (-1, 3, -4) and (1, -3, 4)

$\begin{array}{l}{\mathrm{x}}_{1}=-1,{\mathrm{y}}_{1}=3,{\mathrm{z}}_{1}=-4\\ {\mathrm{x}}_{2}=1,{\mathrm{y}}_{2}=-3,{\mathrm{z}}_{2}=4\end{array}$

Required distance,

$\begin{array}{r}\mathrm{AB}=\sqrt{\left(1+1{\right)}^{2}+\left(-3-3{\right)}^{2}+\left(4+4{\right)}^{2}}\\ =\sqrt{4+36+64}=\sqrt{104}=2\sqrt{26}\end{array}$

(d) Let the given points are A(2, -1,3) and B(-2,1,3).

$\begin{array}{r}{\mathrm{x}}_{1}=2,{\mathrm{y}}_{1}=-1,{\mathrm{z}}_{1}=3\\ {\mathrm{x}}_{2}=-2,{\mathrm{y}}_{2}=1,{\mathrm{z}}_{2}=3\end{array}$

Required distance,

$\begin{array}{r}\mathrm{AB}=\sqrt{\left(-2-2{\right)}^{2}+\left(1+1{\right)}^{2}+\left(3-3{\right)}^{2}}=\sqrt{{4}^{2}+{2}^{2}+0}\\ =\sqrt{16+4}=\sqrt{20}=2\sqrt{5}\end{array}$

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