Question

If the ratio of the roots of the quadrtic equation $x^2+px+q=0$ be equal to the ratio of the roots of $x^2+ix+m=0$, then prove that $p^{2}m=I^{2}q$.

Answer 1

First of all, lets have a look at what we're given. Ratio of roots of x^2 + px + q = 0 and x^2 + rx + m = 0 are equal.

We are assuming that the roots of x^2 + px + q = 0 are α, β while roots of x^2 + rx + m = 0 areγ, δ. So, as we're given,
α / β = γ / δ

By using the sum and product of roots formulae, we can say that

α + β = -p ; αβ = q
γ + δ = -r ; γδ = m

We have to prove that m.p^2 = q.r^2. Now, there can be several approaches to the proof. We'll be telling you two of them.We are given that
α / β = γ / δ (1)
Reciprocating both the sides, we'll get
β / α = δ / γ (2)

Adding (1) and (2), 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.