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Ellipses

Collection of points whose sum of distances from two foci is constant.

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Degenerate Conics

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Vocabulary

Complete the chart.
Word Definition
____________ the distance from the center of the ellipse to the furthest point on the ellipse; represented by the letter  \begin{align*}a\end{align*}
____________ the longest distance from end to end of an ellipse
Semi-minor axis ______________________________________________________________
Ellipse ______________________________________________________________
____________ the two points that the ellipse curves around
Eccentricity ______________________________________________________________
Hyperbola ______________________________________________________________
Degenerate conic ______________________________________________________________

Ellipses

How many foci do ellipses have? ______________

For every point on the ellipse, the sum of the distances to each foci is _______________.

On the following image, label the major, semi-major, and minor axes:

What is the general equation of an ellipse? _______________________

The coefficient  \begin{align*}a\end{align*} always comes from the length of the ____________ axis and the coefficient  \begin{align*}b\end{align*} always comes from the length of the _____________ axis.

How do you find the locations of the two foci? What is the equation? _______________________

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In your own words, describe eccentricity. ________________________________

What two numbers is the eccentricity between? ____________

What happens when the number gets closer to 1? ________________________________

What is the equation for the distance from the center of the ellipse to each directrix? _______________________

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Hyperbolas

What is the equation for a hyperbola that opens side to side? _______________________

Up and down? _______________________

Note:  \begin{align*}a\end{align*} goes with the positive term and  \begin{align*}b\end{align*} goes with the negative term.  It does not matter which constant is larger.

The conjugate axis travels ______________ to the transverse axis through the ____________ and has length _______ .

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How do you measure eccentricity in hyperbolas? ____________________________

What numbers can the eccentricity be for hyperbolas? _________________

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Degenerate Conics

The general equation of a conic is \begin{align*}Ax^2+Bxy+Cy^2+Dx+Ey+F=0\end{align*} . What does this form encompass? ____________________________________________

The three types of degenerate conics are:

  1. singular point , which is of the form: _____________________ . You can think of a singular point as a circle or an ellipse with an infinitely small radius. 
  2. line , which has coefficients  \begin{align*}A=B=C=0\end{align*} in the general equation of a conic.  The remaining portion of the equation is ________________ , which is a line.
  3. degenerate hyperbola , which is of the form:  __________________ .  The result is two intersecting lines that make an “X” shape.  What is the equation for the slopes of the intersecting lines forming the X? ___________
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