Identifying conic sections

See the picture above. Consider a double cone as shown in the picture. Also consider a plane. Let this plane intersect the double cone. The way in which the plane intersects the double cone, would give rise to conic sections. As we can see in the picture the axis of the cone is vertical.

1. Conic sections circles:  If the intersecting plane is perfectly horizontal, that means that if the intersecting plane is perpendicular to the vertical axis, then the conic section thus produced would be a circle.

2. Conic sections ellipse: If the intersecting plane is inclined to the vertical at an angle that is between α to 90 degrees, where α is the angle between the vertical axis and the slant length of the cone, then the conic section thus produced would be an ellipse.

3. Conic sections parabola: If the intersecting plane is inclined to the vertical axis at an angle that is between 0 to α (α is same as described above), then the conic section produced would be a parabola.

4. Conic sections hyperbola: If the intersecting plane is parallel to the vertical axis, then the conic section thus produced would be a hyperbola.

Identifying conic sections:
Conic sections can be identified by two methods. They are, graphical and algebraic.
Graphical method of identifying conic sections:
The graphs of various conic sections are as shown below:

1. Graph of a circle:

The graph of a circle would look like above, it would have a centre and the radius on all sides would be equal.

2.Graph of an ellipse:

The graph of an ellipse can look like any of the two above figures. One is a horizontal ellipse and other is a vertical ellipse. An ellipse would have a major and a minor axis, two vertices and two co-vertices.

3.Graph of a parabola:

A parabolic curve would look like above. It would essentially have a vertex, a focus, a directrix and an axis of symmetry.

A hyperbola can be horizontal or vertical. Algebraic method for identifying conic sections:
In a conic sections practice problem, the equation of the curve should be similar to any of the following general equations:

Equation of the curve	Type of conic section
(x-h)^2 + (y-k)^2 = r^2	Circle
((x-h)/a)^2 + ((y-k)/b)^2 = 1	Ellipse
y-k = 4a(x-h)^2	Parabola
((x-h)/a)^2 - ((y-k)/b)^2 = 1	Hyperbola