Conic_Sections

Unit 10 - Conic Sections
Big ideas:
 * [|Essential Topics: Conic Sections]
 * A geometric view of the conic sections as [|slices of a cone].
 * circle: locus of points equidistance from 1 point
 * ellipse: locus of points whose sum of distances from 2 points (the foci) is constant
 * parabola: all points whose distance to 1 point is the same as the distance to a line (call the directrix)
 * hyperbola: set of points whose difference of distances is constant


 * parent function: x^2 + y^2 = 1

Helpful hints:
 * the y-stretch, x-stretch, and distance to the foci are always a pythagorean triple for (hyperbolas and ellipses)
 * when a conic is in Ax^2 + Cy^2 + Dx + Ey + F = 0, the values of A and C determine the type of conic.
 * 1 coefficient neg and 1 coefficient pos. --> hyperbola
 * when the coefficient of **x^2 is positive** and the y^2 coefficient is negative, the hyperbola opens to the **left and right**
 * when the coefficient of **y^2 is positive** and the x^2 coefficient is negative, the hyperbola opens to the **top and bottom**
 * A and C are positive and equal --> circle
 * 1 coefficient = 0 --> parabola
 * same sign different values ---> ellipse
 * DONT MAKE YOUR LIFE HARD! when you are asked to find the slope of asymptotes for a hyperbola, just use point slope form!!
 * **y = m(x - x ) + y the point you have is (x,y) **
 * to get a conic in Ax^2 + Cy^2 + Dx + Ey + F = 0 form to regular form, **complete the square!**
 * ellipses are internally tangent to the box, hyperbolas are externally tangent.
 * the vertex of a parabola is equidistant from the focus and the directrix
 * all parabola's can be written in the form y = (1/4f) x^2
 * foci are always on the major axis
 * foci are always on the major axis

Sample Problems: (click here for solutions to these problems)
 * 1) In a whisper gallery, people can hear whispers in certain parts of the room. These rooms are usually shaped like ellipses and one can hear whispers at the foci of the ellipse. If a whisper gallery is 68 feet wide and 32 feet wide, how far must 2 people stand apart to hear each other? -Timothy
 * 2) Graph an ellipse with a vertical stretch of 2 and a horizontal stretch of 3 and a hyperbola with the horizontal stretch of 2 and vertical stretch of 3, both centered at the origin. What are the foci of these two conic sections? What is the solution to this system? Be sure to be able to solve it algebraically. -Alexis
 * 3) Two kids are 10 feet apart. Both of these kids are holding onto one end of the same piece of string. A third friend holds on to the string and stands equidistant from both friends with the string being as taught as possible. If the 3rd friend is 7 feet from both of his friends, how close can he get to one of his friends? If he runs around with paint on his feet keeping the string taught, what shape does he paint around his friends? How much data can you get about this shape? Can you get an equation? What would happen if the + sign in this equation turned into a -? Can you draw that shape? -Jonathan (look it's our corner! TAJ)
 * 4) Sketch a graph of ((x-5)/4)^2 + ((y+2)/3)^2 = 1. Show at least 4 points on the graph. For this conic, what is the farthest apart any two points on the curve could be? -Emma (totes continuing down the line...)
 * 5) Complete the square: x^2+y^2+10x+3=0 -Mallory (uh! I ruined the pattern...)
 * 6) Find 2 possible standard equations of an ellipse who has a vertex at (3,3) and has a focus at (7,1). Robert Wilkins
 * 7) What conic shape is defined by the equation: 2x+7y^2+1298+4x^2+78y=4587? --Sadie
 * 8) If you are given that the vertices of the major axis of an ellipse is (-7,-13) and (5,-13) and that a point on the ellipse is 4.2 units from one of the foci, how far is that point from the other focus? -Michael
 * 9) What is the equation of a circle if the diameter has endpoints of (5,4) and (-1,4)? -Bailey
 * 10) What is the equation of an ellipse that has the vertical diameter of 2 and a horizontal diameter of 8 and a center at (1,2)?---Jimmy P. White
 * 11) (x/3)^2 + (y/9)^2 = 1. Determine the x and y intercepts and the foci of this ellipse. -- Priyanka
 * 12) what is the equation of a parabola with a directrix: x=2 and focus (4,4)?
 * 13) Determine the type of graph, sketch it, and state the transformations on the parent function: 4y^2 + 36x^2 = 144. - Caroline
 * 14) What type of conic has the equation y=-5x^2+4x+2y+12? Is the conic horizontal or vertical? - Will
 * 15) Given an ellipse,(x-2)/1)^2+((y-3)/5)^2=1, and a line, x=2, graph a new ellipse using the intersection points as the foci.-Matthew Morgan Bean II