Question

If a, b, c ∈ R and equations ax² + bx + c = 0 and x² + 2x + 9 = 0 have a common root, show that a : b : c = 1 : 2 : 9.

Answer 1

Given equations are : x² + 2x + 9 = 0 ........(i) and 

ax² + bx + c = 0 ........(ii) 

Clearly roots of equation (i) are imaginary since equation (i) and (ii) have a common root, therefore common root must be imaginary and hence both roots will be common. 

Therefore equations (i) and (ii) are identical

∴ a : b : c = 1 : 2 : 9