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Question
If the x-coordinate of a point P on the join of Q (2, 2, 1) and R (5, 1, −2) is 4, then its z-coordinate is
Options
2
1
-1
-2
Solution
- 1
\[\text { Suppose the point P divides the line joining the point Q } \left( 2, 2, 1 \right) \text{ and } R \left( 5, 1, - 2 \right) \text{ in the ratio k: 1 } . \]
\[ \text{ Using the section formula, the coordinates of the point of intersection are given by } \]
\[\left( \frac{k\left( 5 \right) + 2}{k + 1}, \frac{k\left( 1 \right) + 2}{k + 1}, \frac{k\left( - 2 \right) + 1}{k + 1} \right)\]
\[\text { It is given that the X - coordinate of P is 4 } . \]
\[ \Rightarrow \frac{k\left( 5 \right) + 2}{k + 1} = 4\]
\[ \Rightarrow 5k + 2 = 4\left( k + 1 \right)\]
\[ \Rightarrow k = 2\]
\[\text{ Now } , \]
\[Z - \text{ coordinate of P } = \frac{k\left( - 2 \right) + 1}{k + 1}\]
\[ = \frac{2\left( - 2 \right) + 1}{2 + 1} \left [ \text{ Substituting k } = 2 \right]\]
\[ = - 1\]
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