Question

If the roots of the equation x² + 2ax + b = 0 are real and distinct and they differ by at most 2m, then b lies in the interval

• A. [a² – m², a²)
• B. [a² – m², a²]
• C. [a², a² + m²)
• D. None of these

Answer 1

Answer: (A)

[a² – m², a²)

Answer 2

Let α, β be the roots of

x² + 2ax + b = 0

∴ α + β = – 2a and αβ = b

By the given condition |α – β| ≤ 2m ...(1)

∴ (α – β)² ≤ 4m²

⇒ (α + β)² – 4αβ ≤ 4m² ⇒ 4a² – 4b ≤ 4m²

⇒ a² – b ≤ m² ....(2)

Since roots of (1) are real and distinct.

∴ Disc > 0.

∴ 4a² – 4b > 0 ⇒ a² > b

⇒ b < a² ....(3)

From (2) and (3)

a² – m² ≤ b < a² ∴ b ∈ [a² – m², a²]