The plane that passes through the edge of the cone but is not perpendicular to the axis, nor
parallel to the edge, would not be parallel to the y direction that goes
through the center of the cone which creates the parabola. So the
discription is the discription for an ellipse only. Each of the conics has
a different slice through the cone and therefore discription. So make sure that you memorize each of those along with the specific equation. There are special cases of conics that are called degenerate conics that are a point, line, and intersecting lines.
Hyperbolas degenerate into intersecting lines, Circles degenerate into a point, and
parabolas degenerate into a line.
the equation of an ellipse is (x - h)^2 / a^2 + ( y - k)^2/b^2 = 1
if
a = 10
b = 4
then the equation becomes
x^2 / 10^2 + y^2 / 4^2 = 1 is the equation
so 2a = 20
2b = 8
If the horizontal axis is = 20 then 20 = 2a and so a = 10 in the equation that you square ...not 40.
3^2 = 9 not 6
if b > then a then the axis is vertical
you need to be careful with a and b or a^2 and b^2
in the equation and the given information
the length of the major axis = 2a
the length of the minor axis = 2b
so you have to divide both of those in half
to get a and b
and then square them to get a^2 and b^2
the equation of a circle is (x - h)^2 + ( y - k)^2 = r^2
Be careful with 2a and 2b and
a and b or a^2 and b^2
The length of the ellipse major and minor axis is 2a and 2b
so given that 2a is 10 and 2b is 6 then you have to divide both of those in half to get a and b, or 5 and 3 that you put into the equation for the ellipse for a^2 and b^2
so 5^2 and 3^2 go into the formula for the ellipse:
x^2/25 + y^2/9 = 1