Show that the origin lies in the interior of the acute angle between planes $x+2y+2z=9$ and $4x-3y +12z+13=0$. Find the equation of bisector of the acute angle.

Prove that the line joining $(1, 2, 3), (2, 1,-1)$ intersects the line joining $(-1, 3, 1)$ and $(3, 1, 5)$.

Show that the point$\left(-\frac{1}{2},2,0\right)$ is the Circumcentre of the triangle formed by the points $(1, 1, 0), (1,2,1)$ and $(-2, 2,-1)$.

Show that the plane $ax +by+cz+d=0$ divides the line segment joining $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in a ratio$-\frac{a{x}_1+b{y}_1+c{z}_1+d}{a{x}_2+b{y}_2+ c{z}_2+d}$.

A variable plane is at a constant distance $p$ from the origin and meets the axes at $A, B, C$. Through $A, B, C$ planes are drawn parallel to the co-ordinate planes. Show that the locus of their points of intersection is $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{P^2}$.

A variable plane passes through a fixed point $(a, b, c)$ and meets the co-ordinate axes at $A, B, C$. Show that the locus of the point common to the planes drawn through $A, B$ and $C$ parallel to the co-ordinate planes is $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=1$.

The plane $4x + 7y + 4z + 81=0$ is rotated through a right angle about its line of intersection with the plane $5x + 3y + 10z-25=0$. Find the equation of the plane in new position.

The plane $Ix+ my = 0$ is rotated about its line of intersection with the plane $z=0$ through angle measure $\alpha$. Prove that the equation of the plane in new position is $lx+my\pm z\sqrt{l^2+m^2} \tan \alpha =0$.

State which of the following statements are true (T) or false (F):

The line $\frac{x-1}{2}=\frac{y-1}{2}=\frac{z-1}{2}$ passes through the origin.