The coordinates of the point which divides the line segment joining the points (1, –2, 3) and (3, 4, –5) internally in the ratio 2:3 is (x, y, z). Find the value of x+y+z

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Solution

The coordinates of the point P which divides the line segment joining two points $A (x_{1}, y_{1}, z_{1})$ and $B (x_{2}, y_{2}, z_{2})$ internally in the ratio m:n are $(\frac{m x_{2} + n x_{1}}{m + n}, \frac{m y_{2} + n y_{1}}{m + n}, \frac{m z_{2} + n z_{1}}{m + n})$
Let P(x, y, z) be the point which divides line segment joining A(1, –2, 3) and B(3, 4, –5) internally in the ratio 2:3 therefore,
$x = \frac{2 (3) + 3 (1)}{2 + 3} = \frac{9}{5}$
$y = \frac{2 (4) + 3 (- 2)}{2 + 3} = \frac{2}{5}$
$z = \frac{2 (- 5) + 3 (3)}{2 + 3} = \frac{- 1}{5}$
$\Rightarrow x + y + z =$ $\frac{9}{5} + \frac{2}{5} + \frac{- 1}{5}$
$= 2$

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Similar questions

Which of the following statments are correct?

1. The coordinates of the point which divides the line segment joining the points

$(1, - 2, 3)$ and $(3, 4, - 5)$ internally in the ratio $2 : 3$ is $(- 3, - 14, 19)$ .

2. The coordinates of the point which divides the line segment joining the points

$(1, - 2, 3)$ and $(3, 4, - 5)$ externally in the ratio $2 : 3$ is $(\frac{9}{5}, \frac{2}{5}, \frac{- 1}{5})$ .

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