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If \( a, b, c \in R \) and the equations \( a x^{2}+b x+c=0 \) and \( x^{2}+x+1=0 \) have a common root then \( a: b: c \) is equal to(1) \( 1: 1: 1 \)(2) \( 1: 2: 3 \)(3) \( 2: 3: 1 \)(4) \( 3: 2: 1 \) |
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Answer» Correct option is (1) \(1:1:1\) \(x^2 + x + 1=0\) \(x = \frac{-1 \pm \sqrt{-3}}2 = \frac{-1 \pm\sqrt 3 i}2\) Since, roots are imaginary. Given that one root of equation ax2 + bx + c = 0 and x2 + x +1 = 0 is common. Since, root is imaginary so other root is its reciprocal. So, if one root is common then other root must be common. Hence, both given quadratic equations have two common roots. \(\therefore\) Both equation are congruent to each other \(\therefore \frac a1 = \frac b1 = \frac c1\) Hence, \(a = b= c\) \(\therefore a:b:c = 1:1:1\) |
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