let alpha and beta be the roots of the eqⁿ x² – px + r = 0 and alpha /2 and 2beta be the roots of the equation x² - qx + r = 0 . Then the value of r =?

I'm referring to alpha and beta as a,b (it's Easy to solve it)
So,Since a and b are roots of 
We can say a+b=p....say eq1 and ab=r
Also a/2 and 2b are roots of 
So, a/2 +2b=q ..... say eq2 and a/2*2b=ab=r
2(eq1)-eq2=2a -a/2 +2b-2b=2p-q
so 3a/2=2p-q
a=2(2p-q)/3
Substituting in eq1
We get b=p-4p/3+2q/3=(2q-p)/3
So we know ab=r=2(2p-q)(2q-p)/9

 

 

Regards

Arun