Residue fields of the closed points in $Spec(\mathbb{R}[X,Y])$

Question

What are the residue fields of the closed points in $Spec(\mathbb{R}[X,Y])$?

After finding the maximal ideals of $\mathbb{R}[X,Y]$, which are of the form:

1. $\langle X-a,Y-b \rangle$ with $a,b \in \mathbb{R}$

2. $\langle X-a,(Y-b)(Y-\overline{b}) \rangle$ with $a \in \mathbb{R}$ and $b \in \mathbb{C}-\mathbb{R}$

3. $\langle (X-a)(X-\overline{a}),Y-(cX+d) \rangle$ with $a \in \mathbb{C}-\mathbb{R}$ and $c,d \in \mathbb{R}$ such that $d+ca \in \mathbb{C}-\mathbb{R}$

I am trying to compute the quotients

1. $\frac{\mathbb{R}[X,Y]}{\langle X-a,Y-b \rangle}$

2. $\frac{\mathbb{R}[X,Y]}{\langle X-a,(Y-b)(Y-\overline{b}) \rangle}$

3. $\frac{\mathbb{R}[X,Y]}{\langle (X-a)(X-\overline{a}),Y-(cX+d) \rangle}$

Obviously, the quotient 1. is $\mathbb{R}$

For 2., I think $\frac{\mathbb{R}[X,Y]}{\langle X-a,(Y-b)(Y-\overline{b}) \rangle} \simeq \frac{\mathbb{R}[Y]}{\langle (Y-b)(Y-\overline{b}) \rangle} \simeq \mathbb{C}$

I need some help to compute 3.

Thanks!