If for some \[\alpha \] and \[\beta \] in R, the intersection of the
following three places
x + 4y -- 2z = 1
x + 7y -- 5z = b
x + 5y + \[\alpha \]z = 5
is a line in R3, then \[\alpha \] + \[\beta \] is equal to :
Solutions · 1
Official solution
For planes to intersect on a line there should be infinite solution of the
given system of equations.
For infinite solutions
\[\Delta \] = \[\left| {111475−2−5α } \right|\] = 0
\[ \Rightarrow \] 1(7\[\alpha \] + 25) -- 4(\[\alpha \] + 5) -- 2(5 -- 7) = 0
\[ \Rightarrow \] 7\[\alpha \] + 25 -- 4\[\alpha \] -- 20 + 4 = 0
\[ \Rightarrow \] 3\[\alpha \] + 9 = 0
\[ \Rightarrow \] \[\alpha \] = -3
Also \[\Delta \]z = 0
\[ \Rightarrow \] \[\left| {1114751β5 } \right|\] = 0
\[ \Rightarrow \] 1(35 -- 5\[\beta \]) -- 4(5 -- \[\beta \]) + 1(5 -- 7) = 0
\[ \Rightarrow \] 35 - 5\[\beta \] - 20 + 4\[\beta \] - 2 = 0
\[ \Rightarrow \] \[\beta \] = 13
\[ \therefore \] \[\alpha \] + \[\beta \] = -3 + 13 = 10