# Low pass to low pass transformation coefficient?

I am unable to solve this question, 10.10 from GATE IN 2004 (a previous year question paper for an exam targeted at engineering graduates in India.)

So I tried to solve the 10.10 by considering the real frequency shift transformation's general form,

I am not getting any answer close to the options! I wonder what I am doing wrong.

So 1) what is the answer to the question 10.10 shown above?

2) how is this formula for the coefficient derived?

$$e^{-j\omega_0}=\frac{1-\alpha e^{j\hat{\omega}_0}}{e^{j\hat{\omega}_0}-\alpha}\tag{1}$$
For given values of $$\omega_0$$ and $$\hat{\omega}_0$$ you can compute $$\alpha$$ from $$(1)$$:
\begin{align}\alpha&=\frac{1-e^{-j(\omega_0-\hat{\omega}_0)}}{e^{j\hat{\omega}_0}-e^{-j\omega_0}}\\&=\frac{e^{-j(\omega_0-\hat{\omega}_0)/2}}{e^{-j(\omega_0-\hat{\omega}_0)/2}}\cdot \frac{e^{j(\omega_0-\hat{\omega}_0)/2}-e^{-j(\omega_0-\hat{\omega}_0)/2}}{e^{j(\omega_0+\hat{\omega}_0)/2}-e^{-j(\omega_0+\hat{\omega}_0)/2}}\\&=\frac{\sin\left(\frac{\omega_0-\hat{\omega}_0}{2}\right)}{\sin\left(\frac{\omega_0+\hat{\omega}_0}{2}\right)}\tag{2}\end{align}
With $$\omega_0=2\pi\cdot 60/400$$ and $$\hat{\omega}_0=2\pi\cdot 120/400$$ we get from $$(2)$$ the value $$\alpha=-0.45965$$.