Question

If for some $\alpha$ and $\beta$ in R, the intersection of the following three planes
$x+4y-2z=l$
$x+7y-5z=\beta$
$x+5y+\alpha z=5$
is a line in $R^3$, then $\alpha$ + $\beta$ is equal to :

• A. 0
• B. 10
• C. -10
• D. 2

Answer 1

Answer: (B)

For planes to intersect on a line
$\Rightarrow$ there should be infinite solution of the given system of equations
for infinite solutions
$\Delta =\left|\begin{array}{ccc}1& 4& -2\\ 1& 7& -5\\ 1& 5& \alpha \end{array}\right|=0\Rightarrow 3\alpha +9=0\Rightarrow \alpha =-3$
${\Delta }_z=\left|\begin{array}{ccc}1& 4& 1\\ 1& 7& \beta \\ 1& 5& 5\end{array}\right|=0\Rightarrow 13-\beta =0\Rightarrow \beta =13$
Also for $\alpha =-3$ and $b=13{\Delta }_x={\Delta }_y=0$
$\therefore \alpha +\beta =-3+13=10$