Question

If roots of the quadratic equation bx² – 2ax + a = 0 are real and distinct, where a, b belongs to R and b is not equal to 0, then
(A) atleast one root lies in the interval (0, 1).
(B) no root lies in the interval (0, 1).
(C) atleast one root lies in the interval (-1,0)
(D) none of the above.
​

Answer 1

Answer:

Let α, β be the roots of x

2

+2ax+b=0......(1)

⇒α+β=−2a and αβ=b

By hypothesis

∣α−β∣≤2m

⇒(α−β)

2

≤4m

2

⇒(α+β)

2

−4αβ≤4m

2

⇒4a

2

−4b≤4m

2

⇒a

2

−b≤m

2

......(2)

And discriminant of (1) is >0

⇒4a

2

−4b>0

⇒b<a

2

......(3)

From (2) and (3)

a

2

>b≥(a

2

−m

2

)

b∈[a

2

−m

2

,a

2

)