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let a and b are the roots of the equation K(x 2 -x)+ x + 5 = 0 . If K 1 and K 2 are the two values pf Kfor which the roots a and b arec connected by the relation a/b + b/a = 4/5 . Find the value of K 2 /K 1 + K 1 /K 2 .

let a and b are the roots of the equation K(x2-x)+ x + 5 = 0 . If K1 and K2 are the two values pf Kfor which the roots a and b arec connected by the relation a/b + b/a = 4/5 . Find the value of  K2/K1+ K1/K2 .

Grade:12

1 Answers

Arun
25750 Points
6 years ago
Kx^2 + (1 - K)x + 5 = 0 
x^2 + [(1 - K) / K]x + (5 / K) = 0 
And , a and b are the roots of this equation , so 
a + b = -(1 - K) / K = (K - 1) / K ---(#1) 
ab = 5 / K ---(#2) 

Next , from the condition (a/b) + (b/a) = 4/5 , 
(a^2 + b^2) / (ab) = 4/5 
a^2 + b^2 = 4ab / 5 
(a + b)^2 - 2ab = 4ab / 5 ---(#3) 

Substitute (#1) and (#2) into (#3) , 
[(K - 1)^2] / (K^2) - 10 / K = 4 / K 
Multiply both sides by K^2 , 
(K - 1)^2 - 10K = 4K 
K^2 - 2K + 1 - 14K = 0 
K^2 - 16K + 1 = 0 
K = (1/2)(16 ± √252) = 8 ± √63 
k1 = 8 + √63 , k2 = 8 - √63 

Therefore 
(k1/k2)+(k2/k1) 
= (8 + √63) / (8 - √63) + (8 - √63) / (8 + √63) 
= [(8 + √63)^2 + (8 - √63)^2] / [(8 - √63)(8 + √63)] 
= (64 + 16√63 + 63 + 64 - 16√63 + 63) / (64 - 63) 
= 254 / 1 
= 254

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