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Question
Find the points on z-axis which are at a distance \[\sqrt{21}\]from the point (1, 2, 3).
Solution
Let the point be A (0, 0, z).Then,
AP = \[\sqrt{21}\]
\[\Rightarrow \sqrt{\left( 0 - 1 \right)^2 + \left( 0 - 2 \right)^2 + \left( z - 3 \right)^2} = \sqrt{21}\]
\[ \Rightarrow \left( - 1 \right)^2 + \left( - 2 \right)^2 + \left( z - 3 \right)^2 = 21\]
\[ \Rightarrow 1 + 4 + \left( z - 3 \right)^2 = 21\]
\[ \Rightarrow \left( z - 3 \right)^2 = 21 - 5\]
\[ \Rightarrow \left( z - 3 \right)^2 = 16\]
\[ \Rightarrow z - 3 = \pm 4\]
\[ \Rightarrow z - 3 = 4 or z - 3 = - 4\]
\[ \Rightarrow z = 7 or z = - 1\]
Hence, the coordinates of the required point are (0, 0, 7) and (0, 0, −1).
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