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

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The equation of the line passing through the points $A (3, 4, 1)$ and $B (5, 1, 6)$ is given by

$\frac{x - 3}{5 - 3} = \frac{y - 4}{1 - 4} = \frac{z - 1}{6 - 1}$

$\frac{x - 3}{2} = \frac{y - 4}{- 3} = \frac{z - 1}{5} = k$

Any point on this line has coordinates $(2 k + 3, - 3 k + 4, 5 k + 1)$ .

The line crosses the $x y$ -plane where the $z$ -coordinate is zero.

$5 k + 1 = 0 \Rightarrow k = - \frac{1}{5}$

Substituting $k = - \frac{1}{5}$ into the $x$ and $y$ coordinates:

$x = 2 (- \frac{1}{5}) + 3 = \frac{13}{5}$

$y = - 3 (- \frac{1}{5}) + 4 = \frac{23}{5}$

The coordinates of the required point are $(\frac{13}{5}, \frac{23}{5}, 0)$ .

Answer: $(\frac{13}{5}, \frac{23}{5}, 0)$

Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

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