#### 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

2

1

-1

-2

#### Solution

- 1

\[\text { Suppose the point P divides the line joining the point Q } \left( 2, 2, 1 \right) \text{ and } R \left( 5, 1, - 2 \right) \text{ in the ratio k: 1 } . \]

\[ \text{ Using the section formula, the coordinates of the point of intersection are given by } \]

\[\left( \frac{k\left( 5 \right) + 2}{k + 1}, \frac{k\left( 1 \right) + 2}{k + 1}, \frac{k\left( - 2 \right) + 1}{k + 1} \right)\]

\[\text { It is given that the X - coordinate of P is 4 } . \]

\[ \Rightarrow \frac{k\left( 5 \right) + 2}{k + 1} = 4\]

\[ \Rightarrow 5k + 2 = 4\left( k + 1 \right)\]

\[ \Rightarrow k = 2\]

\[\text{ Now } , \]

\[Z - \text{ coordinate of P } = \frac{k\left( - 2 \right) + 1}{k + 1}\]

\[ = \frac{2\left( - 2 \right) + 1}{2 + 1} \left [ \text{ Substituting k } = 2 \right]\]

\[ = - 1\]