What is conic section?

Question

give a simple definition of conic section.

Discuss an application of the conic that you find most interesting.

Include an example of a mathematical equation that models the application How does it apply to the real world?

Answer 1

"conic" comes from "cone"

take 2 cones, turn one upside down, connect them at their apex or their pointed ends

slice through them with a plane, vertically, horizontally and at a slant

you'll get a circle horizontally, hyperbola vertically, ellipses and parabolas at slanted slices

ellipses have real world applications in astronomy

the planets and moons' orbits are close to circles but more elliptical. sometimes circles are just treated as special cases of ellipses where both axes are equal or the foci are the same point

x^2/a^2 + y^2/b^2 = 1 is a general formula for a standard ellipse

a and b are the semi axes of the ellipse

if a is the semi minor axis it's the shortest distance from earth to the moon

b is the longest distance from earth to moon

or distances from earth to the sun

or any planet to the sun

using the earth's center as the center of the ellipse, graphically the origin (0,0)

a parabola is also very interesting. It shows up endlessly, such as in DUI expert testimony

about the level of alcohol in a person's blood after so many hours from driving

it rises at first after drinking, then reaches a peak, the vertex of a downward opening parabola, then decreases on a continuing parabolic curve. if an expert testifies you were drunk at the police station when you took the breath or blood test, but sober when driving, you win your case and get acquitted at trial. all thanks to conic sections and quadratic equations.

Answer 2

Here is a pretty good image that shows conics. Conics are formed by taking different cross sections of cones.

https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3Ac88f77a9cc7914d68baf3dbb272db78f307dbd135f37b70d23fe72a1%2BIMAGE%2BIMAGE.1

The most common and interesting conic used in the real world is a parabola. Below is an example equation of a rocket being launched from the ground. It goes up and then of course, back down. It reached its max height at the vertex of the parabola.

h=-16t² + 128t