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Parabola
Conic:
A conic is the locus (the set of all points that satisfy a given condition) of a point whose undirected distance from a fixed straight line.
The fixed point is called focus (focal point) of a conic section.
The fixed line is called is its directrix.
The constant ratio is the eccentricity of the conic section. It is usually denoted by the letter”e”
A conic is called an ellipse if e<1, a parabola if e = 1 and a hyperbola is e>1.
The line which passes through the focus and is perpendicular to the directrix is called the principal axis (axis of symmetry, major axis)
The chord through the focus parallel to the directrix is the latus rectum.
A point where the conic crosses the principal axis is the vertex of the conic section.
The focus and the dirextrix are “c” units from the vertex; the length of the latus rectum is 4c.
Definition of a parabola:
The parabola is the locus of all points that are the same distance from a line in the plane . the directrix, as fro m a fixed point in the plane, the focus.
Point focus = point directrix
PF= PD
The parabola has one axis of symmetry, which intersects the parabola at its vertex.
The distance from the vertex to the focus is |p|.
The distance from the directrix to the vertex is also |p|.
The standard form of the equation of a parabola with vertex (0, 0):
Te equation of a parabola with vertex (0,0) and focus on the x-axis is y2=4px.
The coordinates of the focus are (p,0).
The equation of the directrix is x= -p
If p>0, the parabola opens right.
If p<0, the parabola opens left.
The equation of a parabola with vertex (0, 0) and focus on the y-axis is x2= 4py.
The coordinate of the focus are (0,p).
The equation of the directrix is y= -p.
If p>0, the parabola opens up.
If p<0, the parabola opens down.
Example:
A parabola has the equation y2=-8x. Sketch the parabola .
V(0, 0)
Focus is on the x axis.
Standard equation is y2=4px.
Here 4p= -8
P=-2
F(-2, 0)
Equation of the directrix is x= -(-2)=2
For a parabola with the axis of symmetry parallel to the y-axis and vertex at ( h, k):
| Axis of symmetry | Focus | Directrix | p>0 | P<0 | Standard form |
| x=h | (h, k+p) | y= k-p | Opens upward | Opens downward | (x-h)2=4p(y-k) |
The general form of the parabola is Ax2+By2+Cx+Dy+E =0 where A=0 or C=0.
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