Question

If c, d are zeroes of x2-10x-11b and a , b are zeroes of x2-10cx-11d then value of a+b+c+d= ?

A) 1210

B)-1

C)2530

D)-11

Answer 1

We know for sure, from the properties of the roots that:
a+b=10ca+b=10c
c+d=10ac+d=10a
a.b=−11da.b=−11d
c.d=−11bc.d=−11b
Which implies
a.c=121a.c=121 and (a+c)=(b+d)/9(a+c)=(b+d)/9

As aa is a root of the first equation,
a2−10ac−11d=0a2−10ac−11d=0
and as cc is a root of the second equation,
c2−10ac−11b=0c2−10ac−11b=0
Adding both equations, we get:
⟹a2+c2−2∗10ac−11(b+d)=0⟹a2+c2−2∗10ac−11(b+d)=0
⟹a2+c2−2∗10ac−2∗ac+2∗ac−11∗9(a+c)=0⟹a2+c2−2∗10ac−2∗ac+2∗ac−11∗9(a+c)=0
⟹(a+c)2−22(ac)−99(a+c)=0⟹(a+c)2−22(ac)−99(a+c)=0
Substituting the value of ac=121ac=121, we get a quadratic equation where (a+c)(a+c)is the variable.
⟹(a+c)2−99(a+c)−2662=0⟹(a+c)2−99(a+c)−2662=0

Let X=(a+c)X=(a+c).

⟹X2−99X−2662=0⟹X2−99X−2662=0
⟹X=(a+c)=−22or121⟹(a+b+c+d)=−220or1210