The circle, ellipse, parabola and hyperbola are all types of conic sections. When a plane intersects a cone at different angles it will produces one of the four curves depending on the slope of the cone and of the plane.
The four conic sections found by intersecting a cone. 
The standard form(s) of each equation (for both the horizontal and vertical orientations) and the important graphing features for each conic section are presented in the table below:
Conic section  Equation  Graph  Characteristics 

Ellipse 
$\frac{\left(xh\right)^2}{a^2}+\frac{\left(yk\right)^2}{b^2}=1$(x−h)2a2+(y−k)2b2=1 


$\frac{\left(xh\right)^2}{b^2}+\frac{\left(yk\right)^2}{a^2}=1$(x−h)2b2+(y−k)2a2=1 


Circle  $\left(xh\right)^2+\left(yk\right)^2=r^2$(x−h)2+(y−k)2=r2 


Hyperbola 
$\frac{\left(xh\right)^2}{a^2}\frac{\left(yk\right)^2}{b^2}=1$(x−h)2a2−(y−k)2b2=1 


$\frac{\left(yk\right)^2}{a^2}\frac{\left(xh\right)^2}{b^2}=1$(y−k)2a2−(x−h)2b2=1 


Parabola 
$y=a\left(xh\right)^2+k$y=a(x−h)2+k



$x=a\left(yk\right)^2+h$x=a(y−k)2+h 

Each conic section has its own standard form that makes it easiest to read off its characteristics. There is, however, a general form for all conic sections:
$Ax^2+By^2+Cx+Dy+E=0$Ax2+By2+Cx+Dy+E=0,
where $A$A, $B$B, $C$C, $D$D and $E$E are real numbers. All conic sections can be represented by this equation, but not all choices of $A$A, $B$B, $C$C, $D$D and $E$E will result in a conic section  as we will see below. This general form is useful for classifying which type of conic section a particular equation represents.
If an equation in this form does represent a conic section, we can determine which one it will be by looking at the values of $A$A and $B$B. The table below outlines this:

Value of $A$A and $B$B  

Parabola 
$A=0$A=0 or $B=0$B=0  
Ellipse 
$A$A and $B$B have the same sign, that is, $A\times B>0$A×B>0  
Circle 
$A$A = $B$B  
Hyperbola  $A$A and $B$B have the opposite sign, that is, $A\times B<0$A×B<0 
To graph a conic in this form it is easiest to rearrange the equation (which often involves completing the square) into a form where it is easier to find its identifying features (centre, vertices, radius etc.).
Below are examples of when $Ax^2+By^2+Cx+Dy+E=0$Ax2+By2+Cx+Dy+E=0 does not represent a conic section:
Use the applet below to investigate the different curves represented by the general conic equation, by varying just the values of $A$A and $B$B.
Note: The values of $C$C, $D$D and $E$E are fixed in the applet at $10$−10, $10$10 and $10$10.
The following equations describe different conic sections. Determine the conic section represented:
Think: Since we already know that these equations represent conics, we can use the values of $A$A and $B$B to determine which conics they are. We can rearrange each equation so that it is in the form $Ax^2+By^2+Cx+Dy+E=0$Ax2+By2+Cx+Dy+E=0 and then read off the values.
Do:
Reflect: We could only use this technique because the questions told us these equations definitely did describe conics. Otherwise we would have had to rearrange the equation to make sure they had a solution. Note also that if $A$A and $B$B have the same sign, it will be an ellipse, but if they are equal as well a circle is a more accurate description.
Determine the curve produced by the equation $3x^2+5y^26x20y+8=0$3x2+5y2−6x−20y+8=0 and draw the graph.
Think: $A$A and $B$B are both positive so this looks like it will be an ellipse. We should first try to rearrange it into the form $\frac{\left(xh\right)^2}{a^2}+\frac{\left(yk\right)^2}{b^2}=1$(x−h)2a2+(y−k)2b2=1 to make sure it is an ellipse. We can then use the values of $a$a, $b$b, $h$h and $k$k to help draw the graph (if it is an ellipse).
Do: Rearrange the equation into the desired form by completing the square
$3x^2+5y^26x20y+8$3x2+5y2−6x−20y+8  $=$=  $0$0 
$3\left(x^22x\right)+5\left(y^24y\right)$3(x2−2x)+5(y2−4y)  $=$=  $8$−8 
$3\left(\left(x1\right)^21\right)+5\left(\left(y2\right)^24\right)$3((x−1)2−1)+5((y−2)2−4)  $=$=  $8$−8 
$3\left(x1\right)^2+5\left(y2\right)^2$3(x−1)2+5(y−2)2  $=$=  $3+208$3+20−8 
$\frac{3\left(x1\right)^2}{15}+\frac{5\left(y2\right)^2}{15}$3(x−1)215+5(y−2)215  $=$=  $\frac{15}{15}$1515 
$\frac{\left(x1\right)^2}{5}+\frac{\left(y2\right)^2}{3}$(x−1)25+(y−2)23  $=$=  $1$1 
The equation describes a ellipse where $a=\sqrt{5}$a=√5 and $b=\sqrt{3}$b=√3, with a centre at $\left(1,2\right)$(1,2).
Notice the vertices at $\left(1\pm\sqrt{5},2\right)$(1±√5,2) and covertices at $\left(1,2\pm\sqrt{3}\right)$(1,2±√3).
What does the following equation represent $2x^23y^212x+30y57=0$2x2−3y2−12x+30y−57=0? If it is a conic section, draw a graph of the conic.
Think: The coefficients of $x^2$x2 and $y^2$y2 have opposite signs, so if this is a conic section then it will be a hyperbola. We can try to rearrange the equation into the more useful hyperbola form $\frac{x^2}{a^2}\frac{y^2}{b^2}=1$x2a2−y2b2=1.
Do:
$2x^23y^212x+30y57$2x2−3y2−12x+30y−57  $=$=  $0$0 
$2\left(x^26x\right)3\left(y^210\right)$2(x2−6x)−3(y2−10)  $=$=  $57$57 
$2\left(\left(x3\right)^29\right)3\left(\left(y5\right)^225\right)$2((x−3)2−9)−3((y−5)2−25)  $=$=  $57$57 
$2\left(x3\right)^2183\left(y5\right)^2+75$2(x−3)2−18−3(y−5)2+75  $=$=  $57$57 
$2\left(x3\right)^23\left(y5\right)^2$2(x−3)2−3(y−5)2  $=$=  $57+1875$57+18−75 
$2\left(x3\right)^23\left(y5\right)^2$2(x−3)2−3(y−5)2  $=$=  $0$0 
$\left(y5\right)^2$(y−5)2  $=$=  $\frac{2}{3}\left(x3\right)^2$23(x−3)2 
$y$y  $=$=  $\pm\sqrt{\frac{2}{3}}\left(x3\right)+5$±√23(x−3)+5 
Reflect: The equation looks like it would be a hyperbola, however when we completed the squares there was no constant term. The result is just a pair of intersecting lines.
Consider the conic section represented by the equation $x^2+y^2=4$x2+y2=4.
Which conic section does this best represent?
Hyperbola
Ellipse
Circle
Parabola
Draw the graph of the conic section.
Consider the conic section represented by the equation $2x^2+8x4y=0$2x2+8x−4y=0.
Which conic section does this best represent?
Circle
Parabola
Hyperbola
Ellipse
Draw the graph of the conic section.
Consider the conic section represented by the equation $25x^2+50x+4y^232y11=0$25x2+50x+4y2−32y−11=0.
Which conic section does this best represent?
Parabola
Ellipse
Circle
Hyperbola
Draw the graph of the conic section.