Question

Let R be the point which divides the line segment joining P(2,1,0) and Q(-1,3,4) into the ratio 1:2 such that PR<PQ. Find the equation of the line passing through R and parallel to the line
x/2=y=z/2

Answer 1

Answer to Question #91552 – Math – Analytic Geometry

$\mathrm{P}(2,1,0),\mathrm{Q}(-1,3,4)$

Let $\mathrm{R}$ be $(\mathrm{x},\mathrm{y},\mathrm{z})$

Using section formula:-

$$x = \frac {m x _ {2} + n x _ {1}}{m + n} = \frac {1 (- 1) + 2 (2)}{1 + 2} = \frac {- 1 + 4}{3} = 1$$ $$y = \frac {m y _ {2} + n y _ {1}}{m + n} = \frac {1 (3) + 2 (1)}{1 + 2} = \frac {3 + 2}{3} = \frac {5}{3}$$ $$z = \frac {m z _ {2} + n z _ {1}}{m + n} = \frac {1 (4) + 2 (0)}{1 + 2} = \frac {4 + 0}{3} = \frac {4}{3}$$

So, $\mathrm{R}\left(1,\frac{5}{3},\frac{4}{3}\right)$

Given parallel line: $\frac{x}{2} = \frac{y}{1} = \frac{z}{2}$

Parallel vector $= \langle a, b, c \rangle = \langle 2, 1, 2 \rangle$

From $R$, $x_{o} = 1$, $y_{o} = \frac{5}{3}$, $z_{o} = \frac{4}{3}$

Thus, equation for required line:

$$\frac {x - x _ {o}}{a} = \frac {y - y _ {o}}{b} = \frac {z - z _ {o}}{c}$$ $$= \frac {x - 1}{2} = \frac {y - \frac {5}{3}}{1} = \frac {z - \frac {4}{3}}{2}$$ $$= \frac {x - 1}{2} = \frac {3 y - 5}{3} = \frac {3 z - 4}{6}$$

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