Question

Algebra: conic sections. Create a picture of a robot with at least 2 types of conic sections. Then explain what the key features are for the two conic section equations

Answer 1

Kindly check below

1) For this problem, let's start out with the Ellipse

So, the key features of an Ellipse are:

a,a = (2) Semi-major axis ( the sum of the semi-major axis yields the longer axis of an ellipse), a=6

b,b =(2) Semi-minor axis (the sum of the semi-minor axis yields the shorter axis of an ellipse), b=4

F1, F2 (Foci)=(2√5,0), (-2√5,0)

A1,A2 =(6,0), (-6,0) Vertices (the endpoints of the semi-major axis)

B1, B2 =(4,0), (-4,0) Co-vertices (the endpoints of the semi-minor axis)

Domain: [-6,6] The set of entries of a function, usually represented by x

Range: [-4,4] The set of outputs of a function, usually represented by y

`(\left(x\right)^(2))/(36)+(\left(y\right)^(2))/(16)=1`

The general formula for an ellipse

2) Now, let's move on to the next conic section. The Hyperbola

In this sketch, we've got the following features:

A_1, A_2: The vertices of a hyperbola are at (5,-1) and (-3,1)

F_1, F_2: Foci of a hyperbola

`\left(1+2√(13),\:1\right),\:\left(1-2√(13),\:1\right)`

In red: The asymptotes, (in this case slant ones. They set the boundaries where the hyperbola does not trespass.

`y=(3\left(x-1\right))/(2)+1,\:\quad \:y=-(3\left(x-1\right))/(2)+1`

Note that these asymptotes have a slope, so we need to express them as linear equations

P: Generic point on the curve.

Domain: Set of entries of a hyperbola. We can tell that this hyperbola has the following Domain:

`D=(-\infty,-3)\cup(5,\infty)`

A hyperbola can be given with a general formula as well:

`(\left(x-1\right)^(2))/(16)-(\left(y-1\right)^(2))/(36)=1`