f the roots ofDU - UD +b .17. If the ratio of the roots of the equatio

1.	f the roots ofDU - UD +b .17. If the ratio of the roots of the equation, Xx + 2x + m = 0, then prove that pm = 1-9.equation, x + px + g = O be equal to ratio of the rooPTC1
Answer» We are assuming that the roots ofx^2 + px + q = 0areα, βwhile roots ofx^2 + lx + m = 0areγ, 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 = qγ + δ = -l;γδ = m We have to prove that m.p^2 = q.l^2. We are given thatα / β = γ / δ ------------(1)Reciprocating both the sides, we'll getβ / 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 = -l, γδ = m, => (-p)^2 /q = (-l)^2 /m=> m.p^2 = q.l^2 Hence Proved.

f the roots ofDU - UD +b .17. If the ratio of the roots of the equation, Xx + 2x + m = 0, then prove that pm = 1-9.equation, x + px + g = O be equal to ratio of the rooPTC1

Answer»

We are assuming that the roots ofx^2 + px + q = 0areα, βwhile roots ofx^2 + lx + m = 0areγ, 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 = qγ + δ = -l;γδ = m

We have to prove that m.p^2 = q.l^2.

We are given thatα / β = γ / δ ------------(1)Reciprocating both the sides, we'll getβ / 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 = -l, γδ = m,

=> (-p)^2 /q = (-l)^2 /m=> m.p^2 = q.l^2

Hence Proved.