JEE PYQ: Quadratic Equation Question 19
Question 19 - 2020 (04 Sep Shift 1)
Let $\alpha, \beta$ be the roots of $x^2 - 3x + p = 0$ and $\gamma, \delta$ be the roots of $x^2 - 6x + q = 0$. If $\alpha, \beta, \gamma, \delta$ form a geometric progression. Then ratio $(2q + p) : (2q - p)$ is:
(1) 3:1
(2) 9:7
(3) 5:3
(4) 33:31
Show Answer
Answer: (2)
Solution
Let $\alpha, \beta, \gamma, \delta$ be in G.P., then $\alpha\delta = \beta\gamma$
$\frac{\alpha}{\beta} = \frac{\gamma}{\delta}$; $\left|\frac{\alpha - \beta}{\alpha + \beta}\right| = \left|\frac{\gamma - \delta}{\gamma + \delta}\right|$
$\frac{\sqrt{9 - 4p}}{3} = \frac{\sqrt{36 - 4q}}{6}$
$\Rightarrow 36 - 16p = 36 - 4q \Rightarrow q = 4p$
$\therefore \frac{2q + p}{2q - p} = \frac{8p + p}{8p - p} = \frac{9p}{7p} = \frac{9}{7}$
BETA
