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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:
- A 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.
- A 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.
- A 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? ___________