Question

# Assertion :If the equation x2+bx+ca=0 and x2+cx+ab=0 have a common root, then their other root will satisfy the equation x2+ax+bc=0 Reason: If the equation x2=bx+ca=0 and x2+cx+ab=0 have a common root, then a+b+c=0

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Both Assertion and Reason are incorrect
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Solution

## The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for AssertionLet α,β be the roots of x2+bx+ca=0 and α,γ be the roots of x2+cx+ab=0, then we have,α2+bα+cα=0 and α2+cα+ab=0Substrating, we have,(b−c)α+a(c−b)=0⇒α=aPutting α=a in equation x2+bx+ca=0, we have a2+ab+ca=0i.e., a+b+c=0 ...(1)Also, we have αβ=ca and αγ=ab⇒β=c and γ=bNow, β+γ=b+c and βγ=bc.Hence, β,γ will be the roots of the equation x2−(b+c)x+bc=0i.e., x2+ax+bc=0. ....[Using (1)]

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