.
1) #1 is not an ellipse. It is a parabola, instead.
2) #2 is an ellipse.
You should apply "completing the squares" procedure to get the ellipse equation in the standard form.
The best place to learn it from are my lessons
- Ellipse definition, canonical equation, characteristic points and elements
- Standard equation of an ellipse
- Identify elements of an ellipse given by its standard equation
- Find the standard equation of an ellipse given by its elements
- General equation of an ellipse
- Transform a general equation of an ellipse to the standard form by completing the square (*)
- Identify elements of an ellipse given by its general equation
in this site.
Your major source is the lesson marked (*) in the list.
Also, you have this free of charge online textbook in ALGEBRA-II in this site
ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this online textbook under the topic
"Conic sections: Ellipses. Definition, major elements and properties. Solved problems".
Save the link to this textbook together with its description
Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson
into your archive and use when it is needed.