Question

If the point $P(a,b,c)$, with reference to $Eq.\left(i\right)$ 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 $6$

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

We have,

$\begin{array}{c}a+8b+7c=0\\ 9a+2b+3c=0\\ 7a+7b+7c=0\end{array}$

Subtracting the last equation from first and multiplying the last equation by $\frac{9}{7}$ and subtracting from the second equation, we obtain

$-6a+b=0and7b+6c=0$

$\begin{array}{c}\therefore \frac{a}{1}+\frac{B}{6}=\frac{C}{-7}=k\\ a=k\\ b=6k\\ c=-7k\end{array}$

As (a,b,c) $k,6k,-7k$ lies on line

$\begin{array}{c}2xy+z=1,wegetk=1\\ thus,7a+b+c=7k+6k-7k=6k=6\end{array}$