Show that the plane ax + by + cz + d = 0 divides the line joining the points(x_{1} , y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) in the ratio

- ax_{1}+ by_{1}+ cz_{1}+ d / ax_{2} + by_{2} + cz_{2} + d .

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Ratio will be same as the ratio of the perpendiculars drawn from these points on the plane. The following figure explains this

Length of perpendicular drawn, in units, from (x_{1}, y_{1}, z_{1}) to the plane

= …(i)

Length of perpendicular drawn, in units, from (x_{2}, y_{2}, z_{2}) to the plane

= …(iii)

Also, the above two values inside the modulus brackets are of opposite signs as the points are on opposite sides of the plane.

Hence ratio =

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