Let `a,b,c,d` be distinct real numbers and a and b are the roots of the quadratic equation `x^2-2cx-5d=0` . If c and d are the roots of the quadratic equation ` x^2-2ax-5b=0` then find the numerical value of `a+b+c+d`.

Answer

3

Explanation

`a+b=2c`.i
`ab=-5x`..ii
`c+d=2a`……iii
`cd=5b`.(iv)
From Eqs (i) and (iii) we get
`a+b+c+d=2(a+c)`
`:.a+c=b+d`………….v
From Eqs (i) and (iii) we get
`b-d=3(c-a)` …………vi
Also `a` is a root of `x^(2)-2cx-5d=0`
`:.a^(2)-2ac-5d=0`.......vii
As c is a root of
`c^(2)-2ac-5b=0`....viii
From eqs vii and viii we get
`a^(2)-c^(2)-5(d-b)=0`
`implies(a+c)(a-c)+5(b-d)=0`
`implies(a+c)(a-c)+15(c-a)=0` [ from Eq. (vi)]
`implies(a-c)(a+c-15)=0`
`:.a+c=15,a-c!=0`
From Eq. (v) we get `b+d=15`
`:.a+b+c+d=a+c+b+d=15+15=30`
`implies` Sum of digits of `a+b|c+d=3+0=3`