Question

If the point $A(2,-9,\lambda )$ and $B(\lambda ,-1,-3)$ lie on the opposite sides of a plane which contain the lines $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$ and $\frac{x}{1}=\frac{7-y}{3}=\frac{z+7}{2}$, then find the largest integral value of $\lambda$.

Answer 1

Answer: 6

$\because \left|\begin{array}{ccc}-1-0& 3-7& -2+7\\ -3& 2& 1\\ 1& 3& 2\end{array}\right|=0$
$\Rightarrow$ lines are intersecting and for intersection points
$-3r_1-1=r_2$ .....(1)
$2r_1+3=7-3r_2$ .....(2)
$-2+r_1=2r_2-7$ .....(3)
$\Rightarrow r_1=-1,r_2=-2$
$\therefore$ intersection point is $(2,1,-3)$
$\therefore$ Equation of plane is $\left|\begin{array}{ccc}x-2& y-1& z+3\\ -3& 2& 1\\ 1& 3& 2\end{array}\right|=0$
$\Rightarrow x+y+z=0$
$\because A(2,-9,\lambda )\text{ and }B(\lambda ,-1,-3)\text{ lie on the opposite side of the of the plane }$
$\Rightarrow (2-9+\lambda )(\lambda -1-3)<0$
$(\lambda -4)(\lambda -7)<0$
$\Rightarrow \text{ Largest integral value is }\lambda =6$