Original problem :
In the $XY$ plane,Let $P_1$ and $P_2$ two points with coordinates $(-1,0)$ and $(1,0)$. $C_1$ is the locus of points whose sum of distances to $P_1$ and $P_2$ is $4$. $C_2$ is the locus of points whose difference of distances to P1 and P2 is $±1$.
With what angle these two places intersecting each other?
Basics through which I think it helps to solve the original problem: $C_1$ and $C_2$ do they belong to which type of curve? Can we have their equation?
I am tottaly stuck here, I cannot translate the assumptions into clear and distinct ideas.
Sum is easy: take a piece of ribbon, length 4 arbitrary units, and fix its endpoints 2 units apart. Take a pencil and use it to draw a curve while keeping the string taut. That's $C_1$. Recognize that shape?
Difference is harder, but if you recognize $C_1$ then you might start looking at conic sections and find the class for $C_2$ there as well.