Question

If every pair from among the equations $x^2+ax+bc=0,x^2+bx+ca=0$, and $x^2+cx+ab=0$ has a common root, then

• A. the sum of the three common roots is $-\frac{(a+b+c)}{2}$
• B. the sum of the three common roots is $2(a+b+c)$
• C. the product of the three common roots is $abc$
• D. the product of the three common roots is $a^2{b}^2{c}^2$

Answer 1

Answer: (A), (C)

The correct options are
A the sum of the three common roots is $-\frac{(a+b+c)}{2}$
C the product of the three common roots is $abc$

Let $x$ & $y$ be the roots of the equation $x^2+ax+bc=0$
$x+y=-a$ ...(1)
$xy=bc$ ...(2)
And $y$ & $z$ be the roots of the equation $x^2+bx+ca=0$
$y+z=-b$ ...(3)
$yz=ac$ ...(4)
And $x$ & $z$ be the roots of the equation $x^2+cx+ab=0$
$x+z=-c$ ...(5)
$xy=ab$ ...(6)
Adding (1),(3) & (5), we get $2x+2y+2z=-a-b-c$

$\Rightarrow x+y+z=\frac{-(a+b+c)}{2}$
Multiplying (2),(4) & (6), we get $x^2{y}^2{z}^2=a^2{b}^2{c}^2$

$\Rightarrow xyz=abc$
Ans: A,C