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# Hyperbola Equations and the Focal Property

## Set of points in a plane whose distances to two foci have the same difference.

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Conic Sections: Hyperbolas

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### Vocabulary

##### Complete the chart.
 Word Definition Hyperbola ________________________________________________________________ ________________ a shape is so large that no circle, no matter how large, can enclose the shape ________________ a line which a curve approaches as the curve and the line approach infinity Perpendicular hyperbola ________________________________________________________________

### Hyperbolas

Hyperbolas two foci, and they can be defined as the set of points in a plane whose distances to these two points have the same difference . So in the picture below, for every pointP\begin{align*}P\end{align*} on the hyperbola, |d2d1|=C\begin{align*}|d_2 - d_1| = C\end{align*} for some constant C\begin{align*}C\end{align*} .

What is the general form for a hyperbola that opens upwards and downwards and whose foci lie on the y\begin{align*}y-\end{align*} axis? ___________________________

What is the shifted equation for a hyperbola that is centered around the point (h,k)\begin{align*}(h,k)\end{align*} opening up and down? ___________________________

Left and right? ___________________________

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Sketch the Hyperbolas:

1. y24(x1)24=9\begin{align*}\frac{y^2}{4} - \frac{(x - 1)^2}{4} = 9\end{align*}
2. (x2)29(y+4)24=1\begin{align*}\frac{(x - 2)^2}{9} - \frac{(y + 4)^2}{4} = 1\end{align*}
3. (x3)29(y+1)216=1\begin{align*}\frac{(x - 3)^2}{9} - \frac{(y + 1)^2}{16} = 1\end{align*}

Identify the equation of the hyperbola using the image:

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#### Asymtotes

For a hyperbola of the form x2a2y2b2=1\begin{align*}\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\end{align*} , which lines are the asymtotes?

_______________________    _______________________

What about the form y2a2x2b2=1\begin{align*}\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\end{align*}?

_______________________    _______________________

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Find the equations of the asymptotes:

1. (x1)219(y+4)21=9\begin{align*}\frac{(x - 1)^2}{1} - \frac{9(y + 4)^2}{1} = 9\end{align*}
2. (y+2)216(x2)21=1\begin{align*}\frac{(y + 2)^2}{16} - \frac{(x - 2)^2}{1} = 1\end{align*}
3. (x4)21(y+1)24=1\begin{align*}\frac{(x - 4)^2}{1} - \frac{(y + 1)^2}{4} = 1\end{align*}
4. y216(x+1)24=1\begin{align*}\frac{y^2}{16} - \frac{(x + 1)^2}{4} = 1\end{align*}
Graph the hyperbolas, give the equation of the asymptotes, use the asymptotes to enhance the accuracy of your graph.
1. (x2)216(y+4)21=1\begin{align*}\frac{(x - 2)^2}{16} - \frac{(y + 4)^2}{1} = 1\end{align*}
2. (x+2)29(y+2)216=1\begin{align*}\frac{(x + 2)^2}{9} - \frac{(y + 2)^2}{16} = 1\end{align*}
3. (x+4)29(y2)24=1\begin{align*}\frac{(x + 4)^2}{9} - \frac{(y - 2)^2}{4} = 1\end{align*}

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