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Grade 12Algebra

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/K_{2 .}

saransh

8 Years agoGrade 12

1 Answer

Arun

8 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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