If x -coordinate of a point P on the line joining the points Q 2 , 2 , 1 and R 5 , 2 , - 2 is 4 , then the y…

Given,

$x$-coordinate of a point $P$ on the line joining the points $Q\left(2,2,1\right)$ and $R\left(5,2,-2\right)$ is $4$,

Now let assume that point $P$ is dividing the point $Q \& R$ in ratio of $\lambda :1$,

So by section formula we get,

$4=\frac{2+\lambda \times 5}{\lambda +1}$

$\Rightarrow 4\left(\lambda +1\right)=2+\lambda \times 5\Rightarrow \lambda =2$

Now finding the $y$ coordinate of point $P$ we get,

$y=\frac{2+2\lambda }{\lambda +1}\Rightarrow y=\frac{6}{3}=2 .........\left(1\right)$

Now finding the $z$ coordinate of point $P$ we get,

$z=\frac{1+\lambda \left(-2\right)}{\lambda +1}=\frac{1-4}{2+1}=\frac{-3}{3}=-1 ...........\left(2\right)$

Now we can see that from equation $\left(1\right) \& \left(2\right)$ we get,

$y-$coordinate $=-2$ ($z$ coordinate of $P$)