Question

If α ≠ β but α² = 5α – 3 and β² = 5β – 3 then the equation whose roots are α/β and β/α is
(a) 3x² – 25x + 3 = 0
(b) x² + 5x – 3 = 0
(c) x² – 5x + 3 = 0
(d) 3x² – 19x + 3 = 0.

Answer 1

(d) : We need the equation whose roots are α/β and β/α which are reciprocal of each other, which means product of roots is (α/β)(β/α) = 1. In our choice (a) and (d) have product of roots 1, so choices (b) and (d)  are out of court. In the problem choice, None of these is not given. If out of four choices only one choice satisfies that product of root is 1 then you select that choice for correct answer. Now for proper choice we proceed as,

α ≠ β, but α² = 5α – 3 and β² = 5β – 3, changing α, β by x 
therefore,  α, β  are roots of x² – 5x + 3= 0 
=> α + β = 5, αβ = 3