Question

If the point $P(a,b,c)$, with reference to $\left(i\right)$, lies on the plane $2x+y+z=1$, then the value of $7a+b+c$ is

• A. $0$
• B. $12$
• C. $7$
• D. $6$

Answer 1

The correct option is D $6$

$\left[\begin{array}{ccc}a& b& c\end{array}\right]\left[\begin{array}{ccc}1& 9& 7\\ 8& 2& 7\\ 7& 3& 7\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$

So, we get the equations

$a+8b+7c=0$ .....$\left(1\right)$

$9a+2b+3c=0$ .....$\left(2\right)$

$7a+7b+7c=0$ or $a+b+c=0$ .....$\left(3\right)$

Let $c=k$ , then

$\Rightarrow b=-\frac{6k}{7},a=-\frac{k}{7}$

Equation of plane $2a+b+c=1$

$\Rightarrow k=-7$

$\Rightarrow a=1,b=6,c=-7$

$\Rightarrow 7a+b+c=6$