why is learning conics important ?

Question

such as hyperbola, ellipse, or parabolas?

Answer 1

Dakayla,

 

Their equations are fairly simple examples of polynomials where their variables are squared so they are useful practice equations for students to learn how to work with polynomials of the second order, which are more complex than first order straight lines. For example,

Parabolas: (y/2)² = fx;  Ellipses: (x/a)² + (y/b)² = 1 ; and Hyperbolas: (x/a)² - (y/b)² = 1, where a,b & f = constants

 

They appear in Nature frequently or have properties that are aesthetically or structurally useful in products made by people.  For example:

 

Parabolas:  Path of a thrown ball or water from your lawn sprinkler dish telescopes to focus incoming parallel light or headlight reflectors to throw out a parallel light beam, McDonalds golden arches or St. Louis' Gateway Arch, construction arches to support a lot of weight as in bridges or the Eiffel Tower in Paris, etc.

 

Ellipses:  Orbits of planets around the Sun or Satellites around the Earth, shapes of ancient amphitheaters, egg-shaped if it isn't lopsided, etc.

 

Hyperbolas:  Nuclear smoke stacks, horse saddles, sand hourglasses in the narrow neck part, architectural shapes, etc.

 

Finally, if you take a solid 3-dimensional cone sitting on its base and slice it in different ways, you get these conic sections.  Slicing it parallel to the base produces a circle which is just a degenerated ellipse with one diameter instead of 2 unequal ones. Slicing somewhat tilted but not cutting the base produces an ellipse. Slicing the cone so that it produces a chord in the base gives you a hyperbola.  And if the sliced chord is the diameter, you get a parabola.

 

Are these enough reasons for you?