The co-ordinates of the point where the line through the points A(3,4,1) and B(5,1,6) crosses the xy-plane

# The co-ordinates of the point where the line through the points $A\left(3,4,1\right)$ and $B\left(5,1,6\right)$ crosses the xy-plane

1. A

$\left(\frac{13}{5},0,\frac{23}{5}\right)$

2. B

$\left(\frac{13}{5},\frac{23}{5},\frac{6}{5}\right)$

3. C

$\left(\frac{13}{5},\frac{23}{5},0\right)$

4. D

$\left(\frac{13}{3},\frac{23}{3},0\right)$

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### Solution:

The xy - plane cuts the line segment joining the points $A\left({x}_{1},{y}_{1},{z}_{1}\right),B\left({x}_{2},{y}_{2},{z}_{2}\right)$ in the ratio $-{z}_{1}:{z}_{2}$

Hence, the xy- plane divides the line segment joining the points  $A\left(3,4,1\right)$and $B\left(5,1,6\right)$ in the ratio $-1:6$

The coordinates of the point which divides the line segment $AB$ in the ratio  $1:6$ externally is $\left(\frac{5-18}{-5},\frac{1-24}{-5},0\right)=\left(\frac{13}{5},\frac{23}{5},0\right)$

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