Relation between Roots and Coefficients for Quadratic
Let α and β b...
Question
Let α and β be the roots of equation px2+qx+r=0,≠0. If p,q,r are in A.P. and 1α+1β=4, then the value of |α−β|is :
A
√349
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B
2√139
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C
619
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D
2√179
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Solution
The correct option is B2√139 Let p,q,r are in AP ⇒2q=p+r....(i)
Given 1α+1β=4⇒=α+βαβ=4
We have α+β=−qp and αβ=rp ⇒−qprp = 4 ⇒q=−4r
From (i), we have 2(−4r)=p+r⇒p=−9r q=−4r
Now |α−β|=√(α+β)2−4αβ
= √(−qp)2−4rp=√q2−4pr|p|
= √16r2+36r2|−9r|=2√139