Question

1. find examples of conic sections in art and architecture. visit web sites to find pictures of artwork or buildings that illustrate one or more conic sections. in your post, give the link as well as your analysis of which conic section is used. examples already given in the lessons for this unit are off limits, so look for something new!
2. one of the biggest challenges in working with conic sections is to look at an equation in standard form and determine which section it describes. explain how you determine which conic section a particular equation describes.
3. what questions do you still have about the unit? ask them here, and your classmates may give you the answers you seek.

Explanation:

1.

The Sydney Opera House has curved shells that resemble parabolic - like shapes (a type of conic section). A link to an image could be from the official Sydney Opera House website (https://www.sydneyoperahouse.com/). The parabolic shape helps in sound reflection and also gives it a unique aesthetic. Parabolas are defined as the set of all points that are equidistant from a fixed point (the focus) and a fixed - line (the directrix).

For a general second - degree equation \(Ax^{2}+Bxy + Cy^{2}+Dx + Ey+F = 0\). If \(B = 0\):

• Circle: \(A = C

eq0\). For example, \(x^{2}+y^{2}=r^{2}\) represents a circle centered at the origin with radius \(r\).

• Ellipse: \(A

eq C\), \(A\gt0\), \(C\gt0\). For example, \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1\) for \(a
eq b\).

• Parabola: Either \(A = 0\) or \(C = 0\) (but not both). For example, \(y=ax^{2}+bx + c\) is a parabola.
• Hyperbola: \(AC\lt0\). For example, \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\).

This is a self - reflection question. One could ask questions like: How do conic sections interact in 3 - D space? Or, What are real - world applications of hyperbolic conic sections other than navigation systems?

Answer:

A link like https://www.sydneyoperahouse.com/ and analysis of parabolic conic - section usage in the Sydney Opera House.

2.