If a point P(α, β, γ) satisfying
( α, β, γ )( 2, 10, 8; 9, 3, 8; 8, 4, 8 ) = ( 0, 0, 0 )
lies on the plane 2 x+4 y+3 z=5, then 6 α+9 β+7 γ is equal to :

Solution

Maths sir · 4 steps

A row vector times a matrix pairs up with the columns, so read the three equations down the columns.

1. Read the columns. Multiplying the row (α β γ) by the matrix gives one entry per column. Each entry must be zero, so you get 2α + 9β + 8γ = 0, 10α + 3β + 4γ = 0 and 8α + 8β + 8γ = 0.
2. Non-zero solutions exist. The determinant is 2(24-32) - 10(72-64) + 8(36-24) = -16 - 80 + 96 = 0. So this homogeneous system has solutions other than (0, 0, 0), and they form a one-parameter family.
3. One parameter. The third equation gives γ = -α - β. Putting this into the first gives -6α + β = 0, so β = 6α and γ = -7α. Check the second equation: 10α + 18α - 28α = 0.

   2α + 4(6α) + 3(-7α) = 5α = 5 ⇒ α = 1

4. So. The plane fixes the point as (1, 6, -7). Then 6α + 9β + 7γ = 6 + 54 - 49 = 11, which is option (b).

The idea to keep. A homogeneous system with zero determinant gives a whole line of solutions. Write every variable in terms of one parameter, then let the extra condition pin it down.

Answer from the NTA final answer key. Solution by PrepWiser.