The given quadratic equation is ax² + bx + c = 0.

Let the roots of the equation be α and β.

We know that α/β = l/m

α = lβ/m

Substituting the value of α in the quadratic equation, we get

a(l²β²/m²) + b(lβ/m) + c = 0

Multiplying the equation by m², we get

a(l²β²) + b(lβm) + cm² = 0

Dividing the equation by aβ²m², we get

l²/a + blm/aβ² + cm²/aβ²m² = 0

Since α and β are the roots of the equation, we can write

α + β = -b/a

αβ = c/a

Substituting the values of α and β in terms of β, we get

(l/m)β + β = -b/a

β = -am/(al + blm)

Substituting the value of β in the equation α = lβ/m, we get

α = -al/(al + blm)


We need to find the value of b²/ac.

We know that α + β = -b/a and αβ = c/a.

Substituting the values of α and β, we get

(-al/(al + blm)) + (-am/(al + blm)) = -b/a

Simplifying the equation, we get

-(al + am)/(al + blm) = -b/a

b/a = (al + am)/(al + blm)

Substituting the value of b/a in the equation αβ = c/a, we get

(-al/(al + blm))(-am/(al + blm)) = c/a

Simplifying the equation, we get

alm² = c(al + blm)

b²/ac = b²a/(ac)

Substituting the values of b/a and c/a, we get

b²/ac = ((al + am)/(al + blm))² / (al/(al + blm))(c/(al + blm))

Simplifying the equation, we get

b²/ac = (l + m)²/(al² + 2ablm + b²lm²)(c/al + blm)

b²/ac = (l + m)²/(a²l²c + 2abclm + b²clm²)

b²/ac = (l + m)²/(ac(l² + 2blm + b²lm²))

b²/ac = (l + m)²/(ac(l + bm)²)

b²/ac = ((l + m)/(l + bm))²

b²/ac = (l² + 2lbm + m²)/(l² + 2lbm + b²lm²)

b²/ac = (l + m)²/(lm)²

b²/ac = (lm)²/(lm)

b²/ac = lm

Therefore, the correct option is (b) lm.