Question

If for some $\alpha$ and $\beta$ in R, the intersection of the following three planes
$x + 4 y - 2 z = l$
$x + 7 y -5 z = \beta$
$x + 5y + \alpha z = 5$
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
$\Delta=\begin{vmatrix}1&4&-2\\ 1&7&-5\\ 1&5&\alpha\end{vmatrix}=0 \Rightarrow 3\alpha+9=0 \Rightarrow \alpha=-3$
$\Delta_{z}=\begin{vmatrix}1&4&1\\ 1&7&\beta\\ 1&5&5\end{vmatrix}=0 \Rightarrow 13-\beta=0 \Rightarrow \beta=13$
Also for $\alpha=-3$ and $b=13\,\Delta_{x}=\Delta_{y}=0$
$\therefore \alpha+\beta=-3+13=10$