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We are given that
α / β = γ / δ (3)
Reciprocating both the sides, we'll get
β / α = δ / γ (4)
Adding (3) and (4), we'll get
=> (α / β) + (β / α) = (γ / δ) + (δ / γ)
=> (α^2 + β^2) /αβ = (γ^2 + δ^2) /γδ
Adding 2 on both the sides,
=> [(α^2 + β^2) /αβ] + 2 = [(γ^2 + δ^2) /γδ] + 2
=> (α^2 + β^2 + 2αβ) /αβ = (γ^2 + δ^2 + 2γδ) /γδ
=> (α + β)^2 /αβ = (γ + δ)^2 /γδ
Now, using α + β = -p, αβ = q, γ + δ = -r, γδ = m,
=> (-p)^2 /q = (-r)^2 /m
=> m.p^2 = q.r^2
Hence proved
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