question_answer

For what value(s) of a, will the two points (1, a, 1) and \[(-3,0,a)\] lie on opposite sides of the plane\[3x+4y-12x+13=0\]?

A) \[a<-1\,or\,a>1/3\]

B)        \[a=0\]only

C) 0 < a < 1         

D)        \[-1<a<1\]

Correct Answer: A

Solution :

[a] \[P\left( 1,a,1 \right)\,\And \,Q\left( -3,0,a \right)\] \[3x+4y-12z+13=0\]              ....(1)             \[\therefore \] Points P & Q lie on opposite sides of the plane (1), then \[\Rightarrow \left\{ 3\left( 1 \right)+4\left( a \right)-12\left( 1 \right)+13 \right\}\times \] \[\left\{ 3\left( -3 \right)+4\left( 0 \right)-12\left( a \right)+13 \right\}<0\] \[\Rightarrow \left( 4a+4 \right)\left( 4-12a \right)<0\] \[\Rightarrow \left( a+1 \right)\left( 3a-1 \right)>0\] \[\Rightarrow \]