Question

If α, β, γ and a, b, c are complex numbers such that `α/a +  β/b + γ/c` = 1 + i and `a/α +  b/β + c/γ` = 0, then the value of `α^2/a^2 +  β^2/b^2 + γ^2/c^2` is equal to ______.

Options

• 0

• – 1

• 2i

• – 2i

Answer 1

If α, β, γ and a, b, c are complex numbers such that `α/a +  β/b + γ/c` = 1 + i and `a/α +  b/β + c/γ` = 0, then the value of `α^2/a^2 +  β^2/b^2 + γ^2/c^2` is equal to 2i.

Explanation:

`α/a +  β/b + γ/c` = 1 + i squaring

`α^2/a^2 +  β^2/b^2 + γ^2/c^2 + 2((αβ)/(ab) + (βγ)/(bc) + (γα)/(ac))` = 2i

or `α^2/a^2 +  β^2/b^2 + γ^2/c^2 + (2αβγ)/(abc)(c/γ + a/α + b/β)` = 2i

∴ `α^2/a^2 +  β^2/b^2 + γ^2/c^2` = 2i