Question

Let $a,b,c$ and $d$ be non-zero numbers such that $x=c$ and $x=d$ are the roots of the equation $x^2+ax+b=0$ and $x=a$ and $x=b$ are the roots of the equation $x^2+cx+d=0.$ Then

• A. $a+b+c+d=-2$
• B. $\frac{a}{c}=\frac{b}{d}$
• C. $ac+bd=5$
• D. $a^2+b^2+c^2+d^2=\frac{2(b^2+d^2)}{a^2+c^2}$

Answer 1

Answer: (A), (B), (C)

$ac+bd=5$

$x^2+ax+b=0$
Roots are $c$ and $d$
Then $c+d=-a$ and $cd=b$

$x^2+cx+d=0$
Roots are $a$ and $b$
Then $a+b=-c$ and $ab=d$

$\Rightarrow a+b+c=0$ and $a+c+d=0$
$\Rightarrow b=d$

$ab=d$ and $cd=b$
$\Rightarrow a=c=1$ and $b=d=-2(\because b,d\ne 0)$