Let α and β are the roots of equation2x2-(p+1)x+(p-1)=0. If α-β=αβ, then what is the value of p ?
1
2
3
-2
Finding the value of p
Sum of roots α and β
⇒α+β=p+12∵sumofrootsofax2+bx+c=0is-ba
Product of roots α and β
⇒αβ=p-12∵productofrootsofax2+bx+c=0isca
It is given that,
(α–β)2=α2β2⇒α2+β2-2αβ=α2β2∵a-b2=a2+b2-2ab⇒α2+β2-2αβ+4αβ=α2β2+4αβ[Adding4αβbothsides]⇒α2+β2+2αβ=α2β2+4αβ⇒(α+β)2–4αβ=α2β2∵a+b2=a2+b2+2ab⇒p+122–4p-12=p-122⇒p+122–p-122=4p-12⇒p+12+p-12p+12-p-12=4p-12∵a2-b2=(a-b)(a+b)⇒p×1=2(p-1)⇒p=2
Hence, option B is correct .
Let α and β be the roots of equation px2+qx+r=0. If p,q,r are in A,P and 1α+1β=4, then the value of |α−β| is