Question

If the points $A(2,-9,\lambda )$ 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 $\frac{x}{1}=\frac{y-7}{-3}=\frac{z+7}{2},$ then the largest integral value of $\lambda$ is

Answer 1

Plane contains the lines $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$ and $\frac{x}{1}=\frac{y-7}{-3}=\frac{z+7}{2}$

$\therefore$ Equation of plane is
$\left|\begin{array}{ccc}x+1& y-3& z+2\\ -3& 2& 1\\ 1& -3& 2\end{array}\right|=0$
$\Rightarrow 7(x+1)+7(y-3)+7(z+2)=0$
$\Rightarrow x+y+z=0$

Since $A(2,-9,\lambda )$ and $B(\lambda ,-1,-3)$ lie on the opposite side of the plane,
$\therefore (2-9+\lambda )(\lambda -1-3)<0$
$\Rightarrow (\lambda -7)(\lambda -4)<0$
$\Rightarrow \lambda \in (4,7)$
Hence, the largest integral value of $\lambda$ is $6.$