Question

If a, b are roots of equation p(x^2-x)+x+5=0 and p_1 and p_2 are two values of p obtained from relation

a/b+ b/a = 4/5. Then, p_1/p_2^2 + p_2/p_1^2= ?

a) 1000

b) 2000

c) 4000

d) None

Solution

Equation: p(x^2-x)+x+5=0=> px^2-(p-1)x+5=0, Roots: a,b

=> a+b = (p-1)/p and ab = 5/p......(1)

a/b+b/a = 4/5

=> a^2+b^2 = 4/5 ab

=> (a+b)^2 - 2ab = 4/5 ab

From equation (1). we will get

=>((p-1)/p)^2 = 14/5 xx 5/p

=> (p-1)^2 = 14p

=> p^2- 16p+1=0

 p_1 + p_2= 16 and p_1p_2=1 ...(2)

We have to find,

p_1/p_2^2 + p_2/p_1^2= (p_1^3+p_2^2)/ (p_1p_2)^2

=> ((p_1+p_2)^3-3p_1p_2(p_1+p_2))/(p_1p_2)^2

=> ((16)^3- 3xx1xx16)/(1)^2

=> 4048

Therefore, option (d) is correct.
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