submanifold and diffeomorphism

Question

Let $a$ be a positive integer and $M=$ {$(w,x,y,z)\in \mathbb{R} | aw^2+x^2+ay^2+z^2=1$}. $$\\$$ a) Using the implicit function theorem or otherwise, show that $M$ is a submanifold of $\mathbb{R}^4$. $$\\$$(b) Consider $p=(0,1,0,0)\in M$.Compute $T_p M$ as a subspace of $T_p \mathbb{R}^4=\mathbb{R}^4$. $$\\$$(c) What is the dimension of $M$? $$\\$$ (d) Consider the function $\phi:M\rightarrow \mathbb{R}^3: (w,x,y,z)\rightarrow (w+x,y,z).$ Show that $\phi$ is a loacal diffeomorphism at $p=(0,1,0,0)$. $$\\$$ Please bear with me if I have asked so many naive questions, I am a undergrad student who did a little bit self study about differential geometry. I have no clue about this question and I guess question (a) is just to calculate the jacobian and is to show it is invertible? (b) I guess is just(0,2,0,0)? I guess (c) is to show it is a sphere inside $\mathbb{R}^4$ and hence $S^3$?.