A rectangular hyperbola passing through $(3,2)$ has its asymptotes parallel to the coordinate axes. If $(1,1)$ is the point of intersection of the two perpendicular tangents of that hyperbola, then its equation is

$E(1,0,0), F(0,2,0), G(0,0,3)$ are respectively the mid-points of the sides $A B, B C, C A$ of $\triangle A B C$. If $a_1, b_1, c_1$ and $a_2, b_2, c_2$ are respectively the direction ratios of $A F$ and $B G$, then $\frac{a_1^2+b_1^2+c_1^2}{a_2^2+b_2^2+c_2^2}=$

If the direction ratios $a, b, c$ of a line $L$ satisfy the relations $a b+b c+c a=0$ and $6 a b+9 b c+8 c a=0$, then the direction cosines of the line $L$ are

The equation of the plane passing through the line of intersection of planes $\pi_1=2 x+6 y+4 z-7=0$, $\pi_2=x-y-2 z-2=03$ and perpendicular to the plane $x+y+2 z-5=0$ is