If z₁, z₂, z₃ C are the vertices of an equilateral triangle, whose centroid is z ₀ , then _ k =1 ^3 ( z _ k - z ₀ )^2 is equal to

Step-by-step solution

aligned & z ₁+ z ₂+ z ₃=3 z ₀ & ( z ₁+ z ₂+ z ₃ )^2=9 z ₀^2 & z ₁^2+ z ₂^2+ z ₃^2+2 ( z ₁^2+ z ₂^2+ z ₃^2 )=9 z ₀^2 & z ₁^2+ z ₂^2+ z ₃^2=3 z ₆^2 aligned aligned & _ k =1 ^3 ( z _ k - z ₀ )^2= ( z ₁- z ₀ )^2+ ( z ₂- z ₀ )^2+ ( z ₃- z ₀ )^2 & = z ₁^2+ z ₂^2+ z ₃^2+3 z ₀^2-2 ( z ₁+ z ₂+ z ₃ ) z ₀ & =6 z ₀^2-6 z ₀^2 & =0 aligned