Question

 The distance between the point (3, 4, 5) and the point where the line \[\frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z - 5}{2}\] meets the plane x + y + z = 17 is

Options

•  1

• 2

•  3

• None of these

   

Answer 1

 3

\[\text{ The coordinates of any point on the given line are of the form } \]
\[\frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z - 5}{2} = \lambda\]
\[ \Rightarrow x = \lambda + 3; y = 2\lambda + 4; z = 2\lambda + 5\]
\[\text{ So, the coordiantes of the point on the given line are } \left( \lambda + 3, 2\lambda + 4, 2\lambda + 5 \right). \text{ This point lies on the plane } ,\]
\[ x + y + z = 17\]
\[ \Rightarrow \lambda + 3 + 2\lambda + 4 + 2\lambda + 5 = 17 \]
\[ \Rightarrow 5\lambda = 5\]
\[ \Rightarrow \lambda = 1\]
\[\text{ So, the coordinates of the point are} \]
\[\left( \lambda + 3, 2\lambda + 4, 2\lambda + 5 \right)\]
\[ = \left( 1 + 3, 2 \left( 1 \right) + 4, 2 \left( 1 \right) + 5 \right)\]
\[ = \left( 4, 6, 7 \right)\]
\[\text{ Now, the distance between the points} \left( 4, 6, 7 \right)\text{ and } \left( 3, 4, 5 \right)\text{ is } \]
\[\sqrt{\left( 3 - 4 \right)^2 + \left( 4 - 6 \right)^2 + \left( 5 - 7 \right)^2}\]
\[ = \sqrt{1 + 4 + 4}\]
\[ = 3 \text{ units } \]

So, the answer is (c).