Question

A perpendicular is drawn from origin to the plane 3x+4y-5z=0. if OP is their perpendicular line segment with point P lying on the plane then determine the coordinates of 'P'​

Answer 1

Planes

Note: We need to change the equation of the given plane to something else, because the given plane itself passes through the origin. We consider the plane to be

$\quad\quad 3x+4y-5z+50=0$.

Solution:

Now given plane is

$\quad\quad 3x+4y-5z+50=0\quad.....(1)$

Here, $OP$ is perpendicular to the plane $(1)$ and $P$ lies on this plane.

The equations of the line $OP$ are given by

$\quad\quad \frac{x}{3}=\frac{y}{4}=\frac{z}{-5}=r$ (say)

Any point on this line is $(3r,\:4r,\:-5r)$

If this be the coordinates of the point $P$, then from $(1)$, we have

$\quad 9r+16r+25r+50=0$

$\Rightarrow 50r=-50$

$\Rightarrow r=-1$

Answer: Thus the coordinates of the point $P$ are $(3,\:4,\:-5)$.

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