If the roots of the equation , ax^2+bx+c=0, are of the form p/p-1 and p+1/p then the value of (a+b+c)^2 is Options: b^2-2ac , b^2-4ac , 2b^2- ac , 4b^2-2ac ans : b^2-4ac Thanks in advance __________________
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Q)If the roots of the equation , ax^2+bx+c=0, are of the form
p/p-1 and p+1/p then the value of (a+b+c)^2 is-
Solution: Product of the roots=[p/p-1] x [p+1/p]=(p-1)/(p+1)=c/a
=>p=(c+a)/(c-a).....(i)
Sum of roots=(p/p-1)+(p+1/p)=-b/a....(ii)
Putting the value of p from (i) to (ii)-
b/a=1-2(c+a/c-a)^2/(c+a/c-a)(c+a-c+a/c-a)
2b(c+a)=-a^2-c^2-6ac
2bc+2ab=-a^2-c^2-4ac-2ac
-2ab-2bc-4ac=a^2+c^2+2ac
a^2+b^2+2ab+2bc+2ac+4ac=0...(iii)
And we know,(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ac...(iv)
From (iii) and (iv),
a^2+b^2+2ab+2bc+2ac+4ac=(a+b+c)^2-4ac+b^2=0
=>(a+b+c)^2=b^2-4ac
Ans:b^2-4ac
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