# If a and b are the roots of x2 – 3x + p = 0 and c, d are the roots x2 – 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q – p) = 17 : 15.

Given that a and b are roots of x2 – 3x + p = 0

a + b = 3 and ab = p ...(i)

It is given that c and d are roots of x2 – 12x + q = 0

c + d = 12 and cd = q...(ii)

Also given that a, b, c, d are in G.P.

Let a, b, c, d be the first four terms of a G.P.

a = a, b = ar c = ar2 d = ar3

Now,

a + b = 3

a + ar = 3

a(1 + r) = 3…(iii)

c + d = 12

ar2 + ar3 = 12

ar2(1 + r) = 12.....(iv)

From (iii) and (iv) we get

3.r2 = 12

r2 = 4

r = ±2

Substituting the value of r in (iii) we get a = 1

b = ar = 2

c = ar2 = 22 = 4

d = ar3 = 23 = 8

ab = p = 2and cd = 4×8 = 32

q + p = 32 + 2 = 34 and q−p = 32−2 = 30

q + p:q−p = 34:30 = 17:15

Hence, proved.

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