Let the arithmetic mean of 1 a a n d 1 b b e 5 16 , a 2 . If is such that a , 4 , b are in A.P., then the equation x 2 - a x + 2 ( - 2 b ) = 0 has: [2026]

Question

Let the arithmetic mean of $\frac{1}{a} and \frac{1}{b} be \frac{5}{16}, a > 2 .$ If $α$ is such that $a, 4, α, b$ are in A.P., then the equation $α x^{2} - a x + 2 (α - 2 b) = 0$ has: [2026]

Smart Achievers · Accepted Answer

(3)

$a = 4 - d, α = 4 + d, b = 4 + 2 d$

$\Rightarrow (4 + d) x^{2} - (4 - d) x + 2 (4 + d - 8 - 4 d) = 0$

$\Rightarrow (4 + d) x^{2} - (4 - d) x + 2 (- 4 - 3 d) = 0$

$Also \frac{\frac{1}{a} + \frac{1}{b}}{2} = \frac{5}{16}$

$\Rightarrow \frac{\frac{1}{4 - d} + \frac{1}{4 + 2 d}}{2} = \frac{5}{16}$

$\Rightarrow d = 2$

$Equation becomes 6 x^{2} - 2 x - 20 = 0$

$3 x^{2} - x - 10 = 0$

$x = 2, - \frac{5}{3}$

Solution

Q.
Let the arithmetic mean of $\frac{1}{a} and \frac{1}{b} be \frac{5}{16}, a > 2 .$ If $α$ is such that $a, 4, α, b$ are in A.P., then the equation $α x^{2} - a x + 2 (α - 2 b) = 0$ has: [2026]

1 both roots in the interval (-2,0)

2 one root in (0,2) and another in (-4,-2)

3 one root in (1,4) and another in (-2,0)

4 complex roots of magnitude less than 2

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