The ratio and the coordinates of the point where the segment joining ( 3 , 5 , 6 ) and ( 4 , 6 , − 3 ) is divided by the YZ-plane are respectively:

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Let the YZ-plane divide the line segment joining the points $(3, 5, 6)$ and $(4, 6, - 3)$ in the ratio $k : 1$ .

The coordinates of the point of division are $(\frac{4 k + 3}{k + 1}, \frac{6 k + 5}{k + 1}, \frac{- 3 k + 6}{k + 1})$ .

Since the point lies on the YZ-plane, its x-coordinate is zero.

$\frac{4 k + 3}{k + 1} = 0 \Rightarrow 4 k + 3 = 0 \Rightarrow k = - \frac{3}{4}$

The negative sign indicates that the division is external. Thus, the ratio is $3 : 4$ externally.

Substituting $k = - \frac{3}{4}$ into the y and z coordinates:

$y = \frac{6 ( - \frac{3}{4} ) + 5}{- \frac{3}{4} + 1} = \frac{- \frac{9}{2} + 5}{\frac{1}{4}} = \frac{\frac{1}{2}}{\frac{1}{4}} = 2$

$z = \frac{- 3 ( - \frac{3}{4} ) + 6}{- \frac{3}{4} + 1} = \frac{\frac{9}{4} + 6}{\frac{1}{4}} = \frac{\frac{33}{4}}{\frac{1}{4}} = 33$

The coordinates of the point are $(0, 2, 33)$ .

Answer: $3 : 4$ externally, $(0, 2, 33)$

Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

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