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प्रश्न
If α, β are the zeroes of the polynomial f(x) = x2 + x – 2, then `(α/β - α/β)`.
If α and β are the zeros of the polynomial f(x) = x2 + x – 2, find the value of `(1/α - 1/β)`.
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उत्तर
By using the relationship between the zeroes of the quadratic polynomial.
We have
Sum of zeroes = `(-("Coefficient of" x))/("Coefficient of" x^2)` and Product of zeroes = `("Constant term")/("Coefficient of" x^2)`
∴ `α + β = (-1)/1` and `αβ = (-2)/1`
⇒ α + β = –1 and αβ = –2
Now, `(1/α - 1/β)^2 = ((β - α)/(αβ))^2`
= `((α + β)^2 - 4αβ)/(αβ)^2` ...[∵ (β – α)2 = (α + β)2 – 4αβ]
= `((-1)^2 - 4(-2))/((-2)^2)` ...[∵ α + β = –1 and αβ = –2]
= `((-1)^2 - 4(-2))/4`
= `9/4`
∵ `(1/α - 1/β)^2 = 9/4`
⇒ `1/α - 1/β = +-3/2`
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