Let's start by recalling the definition of the domain of a function.

Domain
The domain of a function is the set of all possible $x - values$ for which the function is defined.

With this in mind, let's examine the graphs of each separately and then compare their domains. For simplicity, we will consider the graphs of the conic sections centered at the origin. Then, the lines of symmetry of each graph are the $x -$ and $y -$ axis.

Domain of an Ellipse

We know that the graph of an ellipse consists of one smooth curve.

Looking at the graph, the domain of an ellipse is the interval between two real numbers, the leftmost

x -

value and the rightmost

x -

value.

Example : - a \leq x \leq a

Domain of an Hyperbola

We know that the graph of a hyperbola consists of two separate curves.

We see that, for a horizontal hyperbola, there exist $x -$ values where the function is undefined — it does not exist for those values. Whereas, for a vertical hyperbola, the function is defined for every $x -$ value. Therefore, there are two possibilities for the domain of a hyperbola.

If it is a vertical hyperbola, which opens to the left and right, then the domain consists of two intervals.
If it is a horizontal hyperbola, which opens upward and downward, then the domain is all real numbers.

We can give the following domains for these different cases.

Example for Case 1 : Case 2 : x \leq - a or x \geq a All real numbers

Conclusion

Let's summarize what we have found.

Exercise