If the plane $x-2y+3z=0$ is rotated through a right angle about its line of intersection with the plane $2x+3y-4z-5=0$, then the eqn. of the plane in its new position is ?

Question

$\begin {array}{cc} (a)\:28x-17y+9z=0\: & \:(b)\:22x+5y-4z-35=0 \\ \:(c)\:25x+17y+52z-25=0\:& (d)\:x+35y-10z-70=0\end {array} $

Answer 1

Given equations of the given plane are $x-2y+3z=0$..........(i) and

$2x+3y-4z-5=0$.........(ii)

Equation of the plane through the line of intersection of (i) and (ii) is given by

$x-2y+3z+\lambda(2x+3y-4z-5)=0$.

$=x(1+2\lambda)+(y(-2+3\lambda)+z(3-4\lambda)-5\lambda=0$...........(iii)

The equation of the $x-2y+3z=0$ in its new position given by (iii).

But the new position is got by rotating (i) by $90^{\circ}$

$\therefore$ The angle between (i) and (iii) is $90^{\circ}$

$\Rightarrow\:(1,-2,3).(1+2\lambda,\:\:-2+3\lambda,\:\:3-4\lambda)=0$

$\Rightarrow\:1+2\lambda+4-6\lambda+9-12\lambda=0$

$\Rightarrow\:\lambda=\large\frac{7}{8}$

$\therefore$ By substituting $\lambda$ in the required eqn. of the plane (iii) it becomes

$22x+5y-4z+35=0$