Question

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

Answer 1

It is given that a, b are the roots of equation x 2 −3x+p=0 and c, d are the roots of equation x 2 −12x+q=0, where a,b,c,d forms a G.P.

The equation x 2 −3x+p=0 have roots a and b. So,

a+b=3 and ab=p(1)

Similarly, c and d are the roots of equation x 2 −12x+q=0. So,

c+d=12 and cd=q (2)

Since a,b,c,d are in G.P.,

a=x,b=xr,c=x r 2 ,d=x r 3

Substitute values of a and b in (1),

x+xr=3 x(1+r)=3

Substitute values of c and d in (2),

x r 2 +x r 3 =12 x r 2 ( 1+r )=12

Divide the above two results,

x(1+r) x r 2 ( 1+r ) = 3 12 r 2 =4 r=±2

When r=2,

x= 3 1+2 =1

When r=−2,

x= 3 1−2 =−3

Now, 2 cases arise,

Case 1:

When r=2 and x=1.

ab= x 2 r =2

cd= x 2 r 5 =32

Now,

q+p q−p = 32+2 32−2 = 34 30 = 17 15

Case 2:

When r=−2 and x=−3.

ab= x 2 r =−18

cd= x 2 r 5 =−288

Now,

q+p q−p = −288−18 −288+18 = −306 −270 = 17 15

Thus, the ratio from both cases is ( q+p ):( q−p )=17:15.