Find the ratio in which the line segment having the end points

Question:

Find the ratio in which the line segment having the end points A(-1, -3, 4) and B(4, 2, -1) is divided by the xz-plane. Also, find the coordinates of the point of division

Solution:

Let the plane $X Z$ divides the points $A(-1,-3,4)$ and $B(4,2,-1)$ in ratio $k: 1$.

Hence, using section formula $\left(\frac{\mathrm{mx}_{2}+\mathrm{nx}_{1}}{\mathrm{~m}+\mathrm{n}}, \frac{\mathrm{my}_{2}+\mathrm{ny}_{1}}{\mathrm{~m}+\mathrm{n}}, \frac{\mathrm{mz}_{2}+\mathrm{nz}_{1}}{\mathrm{~m}+\mathrm{n}}\right)$, we get

$=\left(\frac{\mathrm{k} \times 4+1 \times-1}{\mathrm{k}+1}, \frac{\mathrm{k} \times 2+1 \times-3}{\mathrm{k}+1}, \frac{\mathrm{k} \times-1+1 \times 4}{\mathrm{k}+1}\right)$

On XZ plane, Y co- ordinate of every point be zero, therefore

$\frac{\mathrm{k} \times 2+1 \times-3}{\mathrm{k}+1}=0$

$2 k-3=0$

$\mathrm{K}=\frac{3}{2}$

The ratio is $3: 2$ in $X Z$ plane which divides the line joined from points $A$ and $B$.