Question

8.
If a and b are the roots of x2 – 3x + p = 0 and c, d are roots of x² - 12x + q = 0,
where a, b, c, d form a GP. Prove that (q + p): (q- p) = 17:15.​

Answer 1

Step-by-step explanation:

Given Quadratic equations are x

2

−3x+p=0 x

2

−12x+q=0 whose roots are a,b and c,d respectively.

Let a=A, b=Ar, c=Ar

2

, d= Ar

3

Then a,b,c,d form aGP with a common ratio r

Now,a+b=3

ab=p

c+d=12

cd=q

a+b=A(1+r)=3

A=

1+r

3

c+d = Ar

2

(1+r)

From these two equations

3r

2

=12

r=±2

Let r=2

A=

1+r

3

=1

a=A=1

b=Ar=2

c= Ar

2

=4

d = Ar

3

=8

p−q

p+q

=

32−2

32+2

=

15

17

Let r=−2

A=

1−r

3

=−3

a=−3

b=Ar=6

c=Ar

2

=−12

d=Ar

3

=24

q−p

q+p

=

−288+18

−288−18

=

−270

−306

=

15

17

Hence, Proved