Question

If roots of the equation $x^2-10cx-11d=0$ are a, b and those of $x^2-10ax-11b=0$ are c, d, then the value of a + b + c + d is (a, b, c and d are distinct numbers)

Answer 1

Given equation $x^2-10cx-11d=0$ with $a,b$ and $x^2-10ax-11b=0$ with $c,d$

$a+b=10c$ and $c+d=10a$

$\Rightarrow (a-c)+(b-d)=10(c-a)$

$\Rightarrow (b-d)=11(c-a)$ ...(i)

Since, $c$ is the root of $x^2-10ax-11b=0$

$\Rightarrow c^2-10ac-11b=0$ ...(ii)

Similarly, $a$ is the root of $x^2-10cx-11d=0$

$\Rightarrow a^2-10ca-11d=0$ ...(iii)

On subtracting equation (iii) from (ii), we get

$(c^2-a^2)=11(b-d)$ ...(iv)

$\therefore (c+a)(c-a)=11\times 11(c-a)$ [from (i)]

$\Rightarrow c+a=121$

$\therefore a+b+c+d=10c+10a$

$=10(c+a)=1210$