Question

If the system of linear equations x + 2ay + az = 0; x + 3by + bz = 0; x + 4cy + cz = 0 has a non-zero solution, then a, b, c ______.

Options

• satisfy a + 2b + 3c = 0

• are in A.P

• are in G.P

• are in H.P

Answer 1

If the system of linear equations x + 2ay + az = 0; x + 3by + bz = 0; x + 4cy + cz = 0 has a non-zero solution, then a, b, c are in H.P.

Explanation:

For homogeneous system of equations to have non zero solution, Δ = 0

`|(1, 2a, a),(1, 3b, b),(1, 4c, c)|` = 0

Applying C₂ `rightarrow` C₂ – 2C₃

`\implies |(1, 0, a),(1, b, b),(1, 2c, c)|` = 0

R₃ `rightarrow` R₃ – R₁, R₂ `rightarrow` R₂ – R₁

`\implies |(1, 0, a),(0, b, b - a),(0, 2c, c - a)|` = 0

`\implies` bc – ab = 2bc – 2ac

`\implies 2/b = 1/a + 1/c`

∴ a, b, c are in Harmonic Progression.