Let $a$ and $b$ be the roots of the equation $x^{2}-10 c x-11 d=0$ and those of $x^{2}-10 a x-11 b=0

Question

Let $a$ and $b$ be the roots of the equation $x^{2}-10 c x-11 d=0$ and those of $x^{2}-10 a x-11 b=0$ are $c, d$ then find the value of $a+b+c+d$, when $a, b, c, d$ are all distinct.

Professor PC Thomas Classes · Answer

Roots of $x^{2}-10 c x-11 d=0$ are $a$ and $b$\ $\Rightarrow \quad a+b=10 c$ and $a b=-11 d$\ Similarly, $c$ and $d$ are the roots of\ $x^{2}-10 a x-11 b=0$\ $\Rightarrow \quad c+d=10 a$ and $c d=-11 b$\ $\Rightarrow \quad a+b+c+d=10(a+c)$ and $a b c d=121 b d$\ $\Rightarrow \quad b+d=9(a+c)$ and $a c=121$\ Also we have, $a^{2}-10 a c-11 d$\ $=0$ and $c^{2}-10 a c-11 b=0$\ $\Rightarrow \quad a^{2}+c^{2}-20 a c-11(b+d)=0$\ $\Rightarrow \quad(a+c)^{2}-22 	imes 121-99(a+c)=0$\ $\Rightarrow \quad a+c=121$ or -22\ For $a+c=-22$\ We get $a=c$\ $	herefore \quad$ Rejecting this value we have\ $a+c=121$\ $	herefore \quad a+b+c+d=10(a+c)=1210$.\