2017 · Q11
If x = t^(1/(t-1)) and y = t^(t/(t-1)), t>0, t ≠ 1, then what is the relation between x and y ?
Elementary MathematicsAlgebra
- Ayˣ = x^(1/y)
- Bx^(1/y) = y^(1/x)
- Cxʸ = yˣ
- Dxʸ = y^(1/x)
Explanation
Write y = t^{t/(t−1)} = (t^{1/(t−1)})ᵗ = xᵗ. Then y·ln x = t^{t/(t−1)}·(ln t)/(t−1) and x·ln y = t^{1/(t−1)}·t(ln t)/(t−1) = t^{t/(t−1)}·(ln t)/(t−1). Since y ln x = x ln y, exponentiating gives xʸ = yˣ: option C. The other options misplace the exponents.