Question

Determine the equation of the plane on which the co - ordinates of the foot of perpendicular drawn from origin O is the point $P(\alpha ,\beta ,\gamma ).$

• A. ${\alpha }^{2}x+{\beta }^{2}y+{\gamma }^{2}z={\alpha }^3+{\beta }^3+{\gamma }^3$
• B. $\alpha x+\beta y+\gamma z={\alpha }^2+{\beta }^2+{\gamma }^2$
• C. ${\alpha }^{2}x+{\beta }^{2}y+{\gamma }^{2}z=\alpha +\beta +\gamma$
• D. none of these

Answer 1

The correct option is B $\alpha x+\beta y+\gamma z={\alpha }^2+{\beta }^2+{\gamma }^2$

Given that $OP$ is normal to plane

So the Drs of normal to the plane is $\alpha ,\beta ,\gamma$

Therefore the equation of plane is $\alpha x+\beta y+\gamma z=k$

The point $(\alpha ,\beta ,\gamma )$ lies on the plane

So we have ${\alpha }^2+{\beta }^2+{\gamma }^2=k$

Therefore the equation of plane is $\alpha x+\beta y+\gamma z={\alpha }^2+{\beta }^2+{\gamma }^2$

Therefore the correct option is $B$