JEE PYQ: Sequence And Series Question 46
Question 46 - 2019 (08 Apr Shift 2)
If three distinct numbers $a, b, c$ are in G.P. and the equations $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$ have a common root, then which one of the following statements is correct?
(1) $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in A.P.
(2) $d, e, f$ are in A.P.
(3) $d, e, f$ are in G.P.
(4) $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in G.P.
Show Answer
Answer: (1)
Solution
Since $a, b, c$ are in G.P. $\Rightarrow b = ar, c = ar^2$
Given equation $ax^2 + 2bx + c = 0$ $\Rightarrow a(x + r)^2 = 0 \Rightarrow x = -r$
$\therefore x = -r$ must satisfy $dx^2 + 2ex + f = 0$
$\Rightarrow dr^2 - 2er + f = 0$
$\Rightarrow \frac{d}{a} + \frac{f}{c} = \frac{d}{a} + \frac{f}{ar^2} = \frac{2e}{b} = \frac{2e}{ar}$
$\therefore \frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in A.P.
BETA
