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Find the distance between the parallel planes x+y−z+4=0

and x+y−z+5=0

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\frac{10}{3 \sqrt{3}}

\frac{10}{9}

\frac{1}{\sqrt{3}}

\frac{3}{10}

answer is C.

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Detailed Solution

Distance between two Parallel Planes: If

ax + by + cz + d 1 = 0 and   ax + by + cz + d 2 = 0

.
Then, the distance between them is

|\frac{d_{2} - d_{1}}{\sqrt{a^{2} + b^{2} + c^{2}}}|

Now, the equation of parallel planes in the question are

\Rightarrow x + y - z + 4 = 0, x + y - z + 5 = 0

Now, on comparing these given parallel planes with the general form of two parallel planes given by

ax + by + cz + d 1 = 0, ax + by + cz + d 2 = 0

we get,

a = 1, b = 1, c = - 1, d 1 = 4, d 2 = 5

Now, on substituting the respective values of the coefficients in the above formula we get,

= |\frac{5 - 4}{\sqrt{1^{2} + 1^{2} + {(- 1)}^{2}}}|

|\frac{1}{\sqrt{1 + 1 + 1}}|

\frac{1}{\sqrt{3}}

As it is a positive value we can remove the mod and write it as

\Rightarrow \frac{1}{\sqrt{3}}

Hence, the correct option is (3).

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