If every pair of equations `x^2+ax+bc=0, x^2+bx+ca=0` and `x^2+cx+ab=0` has a common root then their sum is

Options

A. the sum of the three common roots is `- (1//2)(a + b + c)`
B. the sum of the three common roots is `2(a + b + c)`
C. one of the values of 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

1,3

Explanation

Since each pair has common root, let the roots be ` alpha , beta ` for
Eq(1), ` beta , gamma` for Eq. (2) and ` gamma , alpha ` for Eq (3). Thereofore,
` alpha + beta = - a , alpha beta = bc `
` beta + gamma = - b, beta gamma = ca `
` gamma + alpha = - C, gamma alpha = ab`
Adding, we get
` 2 (alpha + beta +gamma) = - (a + b + c)`
`rArr alpha + beta +gamma = - (1)/(2) (a + b + c)`
Also by multiplying product of roots , we have
` alpha^(2) beta ^(2) gamma^(2) = a^(2) b^(2) c^(2) or alpha beta gamma = abc ` .