Find the coordinates of the point where the line joining
Question:

Find the coordinates of the point where the line joining A(3, 4, 1) and B(5, 1, 6) crosses the xy-plane.

Solution:

Let the plane $X Y$ divides the points $A(3,4,1)$ and $B(5,1,6)$ in ratio $k: 1$.

Hence, using section formula $\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}, \frac{m z_{2}+n z_{1}}{m+n}\right)$, we get

$=\left(\frac{\mathrm{k} \times 5+1 \times 3}{\mathrm{k}+1}, \frac{\mathrm{k} \times 1+1 \times 4}{\mathrm{k}+1}, \frac{\mathrm{k} \times 6+1 \times 1}{\mathrm{k}+1}\right)$

On $X Y$ plane, $Z$ co- ordinate of every point be zero, therefore

$\frac{k \times 6+1 \times 1}{k+1}=0$

$6 k+1=0$

$\mathrm{K}=-\frac{1}{6}$

The ratio is 1:6 externally in $X Z$ plane which divides the line joined from points $A$ and $B$.