Question

On a sheet of paper, draw an ellipse on a coordinate plane centered
at the origin. Make sure the length of the minor axis is different from the length
of the major axis.

a. What is the equation in standard form for your ellipse?
b. What are the coordinates for the center of the ellipse and the two focus
points?
c. What is the semi-major axis length of your ellipse and how do you know this?
d. What is the semi-minor axis length of your ellipse and how do you know this?
e. How might the ellipse you drew be different from the ellipses of others in the
class? How might your ellipse be similar to the ellipses of others in the class?\
Medal and best answer to whoever answers.

Answer 1

The ellipse will have this standard form where a and b are the semi axis lengths. If a > b we call a the major axis and b the minor axis, or vice versa if a < b.

\[\frac{ x^2 }{ a^2 }+\frac{ y^2 }{ b^2 } =1\]



Centre is obviously origin (given) and the focii are given by
\[S(ae,0); S'(-ae,0)\] if a > b, otherwise
\[S(0,be); S'(0,-be)\] if a < b.
Recall that e, the eccentricity of the ellipse (0 < e <1) can be found by using this relationship \[b^2 =a^2(1-e^2)\] if a > b
Otherwise \[a^2 = b^2(1-e^2)\]

Conveniently the focus ae can be found then
\[b^2 =a^2(1-e^2)=a^2 -a^2e^2 = a^2 - (ae)^2 \]
\[(ae)^2 = a^2 - b^2 \rightarrow ae = \sqrt{a^2-b^2}\]
Similarly
\[(be)^2 = b^2 - a^2 \rightarrow be = \sqrt{b^2-a^2}\]