Consider quadratic equations `x^2-ax+b=0 and x^2+px+q=0` If the above equations have one common root and the other roots are reciprocals of each other

Question

Consider quadratic equations `x^2-ax+b=0 and x^2+px+q=0` If the above equations have one common root and the other roots are reciprocals of each other

Consider quadratic equations `x^2-ax+b=0 and x^2+px+q=0` If the above equations have one common root and the other roots are reciprocals of each other, then `(q-b)^2` equals
A. bq(p-a)^(2)`
B. `b(p-a)^(2)`
C. `q(p-a)^(2)`
D. none of these

Monjur Alahi

5 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - A
`(a)` x^(2)-ax+b=0`……..`(i)`
x^(2)-px+q=0`……….`(ii)`
Let the roots of `(i)` be `alpha`, `beta` and that of `(ii)` be `alpha`, `(1)/(beta)`
`:. Alpha+beta=a`, `alphabeta=b`, `alpha+(1)/(beta)=p`, `(alpha)/(beta)=q`
`:.((q-b)^(2))/((p-a)^(2))=(alpha^(2)((1)/(beta)-beta)^(2))/(((1)/(beta)-beta)^(2))`
But `alpha^(2)=alphabeta*(alpha)/(beta)=bq`
`implies (q-b)^(2)=bq(p-a)^(2)`