Question

$\alpha ,\beta$ are the roots of the equation $x^2-3x+a=0$ and $\gamma ,\delta$ are the roots of the equation $x^2-12x+b=0$. If $\alpha ,\beta ,\gamma ,\delta$ form an increasing G.P., then (a,b) =

• A. (3, 12)
• B. (12, 3)
• C. (2, 32)
• D. (4, 16)

Answer 1

The correct option is C (2, 32)

Since $\alpha ,\beta ,\gamma ,\delta$ form an increasing G.P., so $\alpha \delta =\beta \gamma$ where $\alpha <\beta <\gamma <\delta$.
On solving $x^2-3x+a=0,$
we get $x=\frac{1}{2}(3\pm \sqrt{9-4a}).$
Also $\alpha <\beta$.
Hence $\alpha =\frac{1}{2}(3-\sqrt{9-4a}),\beta =\frac{1}{2}(3+\sqrt{9-4a})$
Similarly from $x^2-12x+b=0,$ we get
$\gamma =\frac{1}{2}(12-\sqrt{144-4b}),\delta =\frac{1}{2}(12+\sqrt{144-4b})$
Substituting these values of $\alpha ,\beta ,\gamma ,\delta$ in $\alpha \delta =\beta \gamma$ and simplifying, we get (a,b) = (2,32).