Let $\Omega =\{(x,y)\in\mathbb{R}^2| x^2+y^2<1\}$ be the open unit disc in $\mathbb{R}^2$ with boundary $\partial\Omega$. If $u(x,y)$ is the solution of the Dirichlet problem
$$u_{xx}+u_{yy}=0 ~~~~~~~\text{in}~~ \Omega$$ $$u(x,y)=1-2y^2 ~~~~~\text{on}~~ \partial\Omega,$$
then $u(\frac{1}{2},0)$ is equal to $(A)-1$ $(B)-\frac{1}{4}$ $(C)\frac{1}{4}$ $(D)1$.
I only know separation of variable for finding solution. But I am unable to find $u(x,y)$ completely due to lack of initial or boundary condition. It will be grateful if there is any shortcut method or formula. Thank you all for your time.
Hint: $u(e^{it}) = \cos (2t) = \text { Re } (e^{it})^2.$