Question

l2a, b are the roots of the equation x² – 3x + a = 0 and g, d are the roots of the equation x² – 12x + b = 0. If a, b, g, d form an increasing, GP, then (a, b) is equal to-

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

Answer 1

Answer: (C)

(2, 32)

Answer 2

Since a, b, g, d form an increasing GP.

So ad = bg where a < b < g < d …(i)

We have, x² – 3x + a = 0

Ž x = $\frac{1}{2}$ (3 ±$\sqrt{9 - 4a}$)

Also a < b

Hence, a = $\frac{1}{2}$ (3 – $\sqrt{9 - 4a}$)

and b = $\frac{1}{2}$ (3 + $\sqrt{9 - 4a}$)

Similarly, from x² – 12x + b = 0

Ž x = $\frac{12 \pm \sqrt{144 - 4b}}{2}$

Since, g < d

\ g = $\frac{1}{2}$ (12 – $\sqrt{144 - 4b}$)

and d = $\frac{1}{2}$ (12 + $\sqrt{144 - 4b}$)

Substituting these values of a, b, g, d in Equation. (i)

$\frac{1}{2}$ (3 –$\sqrt{9 - 4a}$) .$\frac{1}{2}$ (12 +$\sqrt{144 - 4b}$)

= $\frac{1}{2}$ (3 + $\sqrt{9 - 4a}$). $\frac{1}{2}$ (12 – $\sqrt{144 - 4b}$).

Only the option (3) (2, 32) satisfy the equation.