Q. 164.8( 5 Votes )

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.

Answer :

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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