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Self-inverse f(x) = (ax + 1)/(x - b) and the value of beta | JEE Advanced

JEE Maths question with a full step-by-step solution.

Question
If f(x)=αx+1x−βf(x) = \dfrac{\alpha x+1}{x-\beta} ∀x∈R−{β}\forall x \in \mathbb{R}-\{\beta\}, with αβ≠1\alpha\beta \ne 1 and α≠1\alpha \ne 1, is a self-inverse function such that
f(4)4=f(12)12=f(1+β1−α),\frac{f(4)}{4} = \frac{f(12)}{12} = f\left(\frac{1+\beta}{1-\alpha}\right),
 then "β\beta" is equal to
Solution
Answer: 8 (± 0.01)
Step 1: Solving y=αx+1x−βy = \dfrac{\alpha x+1}{x-\beta} for xx,
y(x−β)=αx+1  ⟹  x(y−α)=1+βy  ⟹  f−1(y)=βy+1y−αy\left(x-\beta\right) = \alpha x+1 \;\Longrightarrow\; x\left(y-\alpha\right) = 1+\beta y \;\Longrightarrow\; f^{-1}(y) = \frac{\beta y+1}{y-\alpha}
 f−1=ff^{-1} = f means the two expressions agree for every yy, and
(βy+1)(y−β)−(αy+1)(y−α)=(β−α)[y2−(α+β)y−1]\left(\beta y+1\right)\left(y-\beta\right)-\left(\alpha y+1\right)\left(y-\alpha\right) = \left(\beta-\alpha\right)\left[y^2-\left(\alpha+\beta\right)y-1\right]
 vanishes for every yy only when
α=β\alpha = \beta
 Step 2: With α=β\alpha = \beta, put x0=1+β1−βx_0 = \dfrac{1+\beta}{1-\beta}, which is defined since β=α≠1\beta = \alpha \ne 1.
f(x0)=βx0+1x0−β=β(1+β)1−β+11+β1−β−β=β+β2+1−β1+β−β+β2=β2+1β2+1=1f\left(x_0\right) = \frac{\beta x_0+1}{x_0-\beta} = \frac{\dfrac{\beta\left(1+\beta\right)}{1-\beta}+1}{\dfrac{1+\beta}{1-\beta}-\beta} = \frac{\beta+\beta^2+1-\beta}{1+\beta-\beta+\beta^2} = \frac{\beta^2+1}{\beta^2+1} = 1
 (the denominator β2+11−β\dfrac{\beta^2+1}{1-\beta} is never 00, so x0≠βx_0 \ne \beta). Step 3:
f(4)4=1  ⟹  f(4)=4,f(12)12=1  ⟹  f(12)=12\frac{f(4)}{4} = 1 \;\Longrightarrow\; f(4) = 4 ,\qquad \frac{f(12)}{12} = 1 \;\Longrightarrow\; f(12) = 12
 so 44 and 1212 are fixed points of ff, and
f(x)=x  ⟹  βx+1x−β=x  ⟹  βx+1=x2−βx  ⟹  x2−2βx−1=0f(x) = x \;\Longrightarrow\; \frac{\beta x+1}{x-\beta} = x \;\Longrightarrow\; \beta x+1 = x^2-\beta x \;\Longrightarrow\; x^2-2\beta x-1 = 0
 Step 4: The two roots of this quadratic are 44 and 1212, so by the sum of the roots,
4+12=2β⟹β=84+12 = 2\beta \quad\Longrightarrow\quad \beta = 8
 Answer: 88
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