Question

Consider the planes $3x-6y-2z=15$ and $2x+y-2z=5$.

Statement-$1$: The parametric equation of the line of intersection of the given planes are $x=3+14t$, $y=1+2t,$ and $z=15t$.
Statement-$2$: The vector $14\hat{i}+2\hat{j}+15\hat{k}$ is parallel to the line of intersection of given planes

• A. Both statement-$1$ a statement-$2$ are corrrect and statement-$2$ is the correct explanation of statement-$1$
• B. Both statement-$1$ a statement-$2$ are corrrect and statement-$2$ is not the correct explanation of statement-$1$
• C. Statement-$1$ is true a statement-$2$ is false
• D. Statement-$1$ is false a statement-$2$ is true

Answer 1

The correct option is D Statement-$1$ is false a statement-$2$ is true

Subtracting the two equations gives $x-7y=10,\Rightarrow x=3,y=-1$ is a possible solution, which gives $z=0$.

This implies that $(3,-1,0)$ is a point on the line.

Similarly take $z=5$, we have $3x-6y=25$ and $2x+y=15$.
Solving these two simultaneously, we get $15x=115,\Rightarrow x=\frac{23}{3}$
Also, $y=\frac{-1}{3}$
Thus the direction vector becomes $(\frac{14}{3},\frac{2}{3},5)$ i.e. $(14,2,15)$
Thus statement $1$ is false but $2$ is true.