Let α, β, γ and a, b, c be complex numbers satisfying

(α)/(a)+(β)/(b)+(γ)/(c)=1+i
and
(a)/(α)+(b)/(β)+(c)/(γ)=0.

Then the value of

(α²)/(a²)+(β²)/(b²)+(γ²)/(c²)

is

Question

Let α, β, γ and a, b, c be complex numbers satisfying

(α)/(a)+(β)/(b)+(γ)/(c)=1+i
 and
(a)/(α)+(b)/(β)+(c)/(γ)=0.

Then the value of

(α²)/(a²)+(β²)/(b²)+(γ²)/(c²)

is

Accepted Answer

2i. Let x=alpha/a, y=beta/b, z=gamma/c. Then x+y+z=1+i and 1/x+1/y+1/z=0 gives xy+yz+zx=0. So x^2+y^2+z^2 = (x+y+z)^2 - 2(xy+yz+zx) = (1+i)^2 = 2i.

Solution