Since a and b are the roots of x2 – 3x + p = 0 and c, d are roots of
x2 – 12x + q = 0
Then, a + b = 3, ab = p, c + d = 12, cd = q
Also, a, b, c, d forms a G.P., then if a is first term and r is a common ratio, then
b = ar, c = ar2, d = ar3
a + b = 3 ⇒ a + ar = 3 ⇒ a (1 + r) = 3 ...(i)
c + d = 12 ⇒ ar2+ar3 = 12 ⇒ ar2 (1 + r) = 12 ...(ii)
Dividing (ii) by (i), we get, r2 = 4 ...(iii)
Now, ab = a(ar) = a2r = p ...(iv)
cd = (ar2)(ar3) = a2r5 = q ...(v)
Dividing (v) by (iv), we get r4 =

p

q

⇒ (4)2 =

q

p

[using (iii)]
⇒

q

p

=

16

1

Applying componendo and dividendo, we get

q+p

q−p

=

17

15

i.e.,
(q + p) : (q – p) = 17 : 15