Question
Question: If the roots of equation x<sup>2</sup> – ax + b = 0 are real and differ by a quantity less than 1. T...
If the roots of equation x2 – ax + b = 0 are real and differ by a quantity less than 1. Then –
A
4a2−1< b <4a2
B
5a2−1< b <5a2
C
6a2−1< b <2a2
D
None of these
Answer
4a2−1< b <4a2
Explanation
Solution
Roots are real and unequal a2 – 4b > 0
⇒ b < 4a2
From question | α – β | < 1, α, β are roots
(α – β)2 = (α + β)2 – 4αβ = a2 – 4b
⇒ | α – β |2 = a2 – 4b
But | α – β | < 1 ⇒ | α – β |2 < 1
⇒ a2 – 4b < 1
⇒ a2 – 1 < 4b ⇒ 4a2−1< b
∴4a2−1 < b < 4a2