Sequences & Series
AP-AM Condition — Quadratic Root Interval
nta_pyq_2026_jan
Grade None
Question:
Let the arithmetic mean of $\dfrac{1}{a}$ and $\dfrac{1}{b}$ be $\dfrac{5}{16}$, $a>2$. If $\alpha$ is such that $a,4,\alpha,b$ are in A.P., then the equation $\alpha x^2-\alpha x+2(\alpha-2b)=0$ has:
one root in (0,2) and another in (-4,-2)
one root in (1,4) and another in (-2,0)
both roots in the interval (-2,0)
complex roots of magnitude less than 2
Step-by-Step Solution
Key Concept: AM: $\tfrac{1}{a}+\tfrac{1}{b}=\tfrac{5}{8}$. AP: $d=4-a=\alpha-4=b-\alpha$; so $\alpha=4+d$, $b=4+2d$, $a=4-d$. Substituting: $5d^2-6d-8=0\Rightarrow d=2$ (rejected, $a=2\not>2$) or $d=-\tfrac{4}{5}$.
Roots $\tfrac{1+\sqrt{5}}{2}\in(1,4)$ and $\tfrac{1-\sqrt{5}}{2}\in(-2,0)$.
Correct Answer: 2