Find a & b of Ellipse from Cone

[femas]

Hi,

If the cone is cut with a plane such that an ellipse has been formed. Let's say the major axis is 'a' and the minor axis is 'b'.

Is there a way to find a and b from the geometry instead of getting them from the quadratic equation.

 

[Eynstone]

The inclination of the plane would give the ratio a:b & its distance from the vertex of the cone would give a. I would prefer to do this trigonometrically.

 

[femas]

Thank you ... But can you give me an example.

 

[slider142]

The distance of the plane from the vertex of the cone as measured along the axis of the cone and the slope of the cone determines the semi-minor axis of the ellipse, while the inclination of the plane together with that distance determines the semi-major axis.

 

[blue_raver22]

Thank you for your question. Yes, there is a way to find the values of a and b from the geometry of the ellipse formed from the cone. First, let's define some terms:

- The cone has a circular base with radius r and height h.
- The ellipse formed from the cone has a major axis of length a and a minor axis of length b.
- The center of the ellipse is located at a distance c from the base of the cone, where c is the distance from the vertex of the cone to the center of the ellipse.

To find a and b, we can use the following equations:

a = 2c + 2r
b = 2c

These equations can be derived from the fact that the distance from the center of the ellipse to any point on the ellipse is equal to the distance from the vertex of the cone to that same point. This is known as the property of tangents for a cone and an ellipse.

So, by finding the values of c, r, and h from the geometry of the cone, we can calculate the values of a and b for the ellipse formed from it. This method is more accurate and reliable than using the quadratic equation, as it takes into account the exact geometry of the cone and the resulting ellipse.

I hope this helps answer your question. Let me know if you have any further inquiries.