The ratio in which the plane divides the line joining the points and .

Engineering

Mathematics

Plane and Its Different Forms

Equation of Straight Line in 3D

Distance from a Plane

Question

The ratio in which the plane $\vec{r} \cdot (\vec{i}, - 2 \vec{j}, + 3 \vec{k}) = 17$ divides the line joining the points $−2 \vec{i}, + 4 \vec{j}, + 7 \vec{k}$ and $3 \vec{i}, - 5 \vec{j}, + 8 \vec{k}$ .

1 : 5

1 : 10

3 : 5

3 : 10

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Let the plane

\vec{r} \cdot (\vec{i}, - 2 \vec{j}, + 3 \vec{k}) = 17

divided the line joining the points

- 2 \vec{i}, + 4 \vec{j}, + 7 \vec{k}

and

3 \vec{i}, - 5 \vec{j}, + 8 \vec{k}

in the ratio t:1 at point P.

Therefore, point P is

\frac{3 t - 2}{t + 1} \vec{i} +, \frac{- 5 t + 4}{t + 1} \vec{j} + \frac{8 t + 7}{t + 1} \vec{k}

This lies on the given plane

∴ \frac{3 t - 2}{t + 1} (1) +, \frac{- 5 t + 4}{t + 1} (- 2) + \frac{8 t + 7}{t + 1} (3) = 17

Solving, we get

t = \frac{3}{10}