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Circles and Parabolas
Transcript of Circles and Parabolas
Alice W., Meen, Nick, Woody
Conic (Conic section) is the intersection of a plane and a double-napped cone.
Parabola = the set of all points (x,y) in a plane that are equidistant from a fixed line, the directrix, and a fixed point, the focus, not on the line.
Standard Form Equation
General Form Equation
Vertical axis: y = ax^2 + bx + c
Horizontal axis: x = ay^2 + by + c
Parabolas in real life
Picture 1: Parabola receiver- when a person whisper at the focus of one of the parabola reflector, it could be hear on the other parabola reflector.
The degenerate form of parabola is a straight line!
because in order to make a parabola, we have to have a cutting plane angle equal to generator angle.
Thus, we will get a straight line if we cut through vertex.
Directrix - The line that is perpendicular to the axis of symmetry.
- directrix = (y-p)
Degenerate conic - The resulting figure of when the plane does pass through the vertex.
Vertex = the midpoint between the focus and the directrix.
Axis of the parabola = the line passing through the focus and the vertex.
Vertical axis; directrix (x-h)^2 = 4p(y-k)
Horizontal axis; directrix (y-k)^2 = 4p(x-h)
Focus = the fixed point inside of a parabola
Picture 2: parabola reflector has the property that when the light source is placed at the focus point, the light ray will reflect from the mirror as ray parallel to the axis.
- (h,k) is the vertex
- The focus lies on the axis p units (directed distance) from the vertex
- h determines the horizontal shift
- k determines the vertical shift
- "a" represents if the parabola open upward or downward
- x = -b/2a
axis of symmetry
Reflective property of a parabola
- The tangent line to a parabola at a point P makes equal angles with the following two lines
1. The line passing through P and the focus
2. The axis of parabola
Parabolas can be found in many places in our daily lives, we just need to notice them.