Relation between Roots and Coefficients for Quadratic
Let α, β are ...
Question
Let α,β are the roots of the equation ax2+bx+c=0,(b≠0) and α4,β4 are the roots of the equation lx2+mx+n=0. If α,β are real and distinct, then the roots of the equation a2lx2−4aclx+2c2l+a2m=0 are
A
always real
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B
opposite in sign
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C
same in sign
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D
always imaginary
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Solution
The correct option is B opposite in sign α+β=−ba,αβ=ca α4+β4=−ml,α4β4=nl
Now, α4+β4=−ml ⇒(α2+β2)2−2α2β2=−ml ⇒((α+β)2−2αβ)2−2α2β2=−ml ⇒(b2a2−2ca)2−2c2a2=−ml ⇒(b2a2)2−4(b2a2)(ca)+2(c2a2)+ml=0 ⇒a2(b2a2)2−4ac(b2a2)+2c2+a2ml=0 ⇒a2l(b2a2)2−4acl(b2a2)+2c2l+a2m=0
Therefore, b2a2 is a root of the equation a2lx2−4aclx+2c2l+a2m=0
Also, b2a2>0
Thus, the equation a2lx2−4aclx+2c2l+a2m=0 has a positive real root.
Let γ be the other root of this equation.
Then, γ+b2a2=4acla2l ⇒γ=4ca−b2a2=4ac−b2a2 ⇒γ<0
Hence, roots are always real and opposite in sign.