Explain identifying conic sections, Mathematics

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Explain Identifying Conic Sections

The graph of a quadratic equation in the variables x and y, like this one,
x2 + 3y2 + 6y = -4, is a conic sections. There are three kinds of conic section:
• hyperbola (a pair of bent lines)
• parabola ( a single bent line)
• ellipse ( a bent circle).

You can identify these three types just by looking at the equation.
Hyperbolas have equations with both an x2 and a y2 term, and there terms have opposite signs (when written on the same side of the equation). Here are three examples:
x2 - y2 = 1
-2x2 + x + y2 = 0
x2 + y =2 + y2
And this is what the graph of a typical hyperbola looks like:


Note that if the squared terms in the third example were moved to the same side of the equation, they would have opposite signs.

2362_Hyperbolas.png

Ellipses also have equations with both an x2 and a y2 term, and these terms have the same sign. Here are three examples of ellipses:
x2 + y2 = 1
-2x2 + x - y2 = 0
x2 + y = 2 - y2
Here is the graph of the first example:

724_Ellipses.png

As you can see, some ellipses- the ones that aren't unevenly scaled- are just circles! Parabolas have equations with only one variable (x or y) squared, but not both.
y = x2
-2x - y2 = 0
x2 + y = 2- y
Graphs of parabolas look something like this:

Note: If the equation has an "xy" term, then you have a rotated conic section. Most calculus courses avoid this somewhat complicated issue, and deal only with non-rotated conics.

1594_Parabolas.png


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