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# Conic Sections and Dandelin Spheres

## Balls used to define conic sections and prove focal properties.

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Conic Sections and Dendelin Spheres

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### Vocabulary

##### Complete the definitions.
 Word Definition Dandelin Spheres __________________________________________________________________ Tangent __________________________________________________________________ Ellipse __________________________________________________________________ Parabola __________________________________________________________________ Hyperbola __________________________________________________________________

### Conic Sections

##### Dandelin Spheres

In your own words, describe how Dandelin used spheres to find the foci and prove the focal property:

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In your own words, describe how Morton used Dandelin Spheres to prove the focal property for parabolas:

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Finally, in your own words, describe how Dandelin Spheres prove the focal property for hyperbolas:

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1. If two tangents are drawn from a single point to a sphere, what can you say about the line segments formed
2. What is the line that results from the intersection between the cutting plane and the plane that contains the circle of contact between the sphere and cone?
3. What is defined by the point where the sphere intersects the cutting plane?
4. How do the tangents relate to a radius of a sphere?
5. Identify the parts listed on the diagram as specified below:

1. Directrix Line - Small Sphere
2. Directrix Line - Large Sphere
3. Focus - Small Sphere
4. Focus - Large Sphere
5. Vertex - Small Sphere
6. Vertex - Large Sphere
7. Directrix Plane - Small Sphere
8. Directrix Plane - Large Sphere
9. Cutting Plane
10. What conic section is illustrated here?

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#### General Forms of Conic Sections

##### Complete the chart.
 Conic Section Properties Equation Ellipse _____________________________________________________ ____________________ Parabola _____________________________________________________ ____________________ Hyperbola _____________________________________________________ ____________________

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How can we look at a degree-2 polynomial equation and determine which conic section it depicts?

Ax2+By2+Cxy+Dx+Ey+F=0\begin{align*}Ax^2 + By^2 + C xy + Dx + Ey + F=0\end{align*}

If A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} differ in sign, the equation is a ____________________.

if A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} are both positive/negative, the conic section is a ____________________.

if A\begin{align*}A\end{align*} or B\begin{align*}B\end{align*} equals zero the equation is a ____________________.

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Identify the conic that is represented by the equation.

1. (x5)24+(y4)24=1\begin{align*} \frac{(x - 5)^2}{4} + \frac{(y - 4)^2}{4} = 1\end{align*}
2. (x3)2+y22=1\begin{align*} (x - 3)^2 + y - 2^2 = 1\end{align*}
3. x25xy24y+16=0\begin{align*} x^2 - 5x - y^2 - 4y + 16 = 0\end{align*}
Convert to standard form:
1. 9x2+4y236x+64y+256=0\begin{align*}9x^2 + 4y^2 - 36x + 64y + 256 = 0\end{align*}
2. 9x2+y2+90x8y+232=0\begin{align*}9x^2 + y^2 + 90x - 8y + 232 = 0\end{align*}
3. 9x29y2+162x+162y81=0\begin{align*}9x^2 - 9y^2 + 162x + 162y - 81 = 0\end{align*}

Identify and Graph the following:

1. (x1)24=(y)216=1\begin{align*}\frac{(x - 1)^2}{4} = \frac {(y)^2}{16} = 1 \end{align*}
2. x=2(y+2)21\begin{align*}x = 2(y + 2)^2 - 1\end{align*}
3. x=(y2)2+4\begin{align*}x = -(y - 2)^2 + 4\end{align*}

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