Parabola
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CONIC SECTIONS
What are conic sections?
A conic section or a conic, as it is sometimes referred to, can be defined as the locus of a point that moves in a plane, such that the ratio of its distance from a fixed point to its distance from a fixed line is always constant. This constant ratio is referred to as the eccentricity of the conic, and is represented by e. The fixed point is called the focus of the conic and the line called the directrix of the conic. Conic sections are called so because they can be obtained as sections of a right circular cone, by a plane in various positions.
NOTE: a circle is also a conic section. It can be considered as a limiting case of an ellipse.
There are three types of conic sections, depending upon the value of e.
|
Ellipse |
e < 1 |
|
Parabola |
e = 1 |
|
Hyperbola |
e 1 |
PARABOLA
According to the above definition of a conic section, a parabola is defined as the locus of a point, which moves such that its distance from a fixed point is equal to its distance from a fixed line. (See Diagram)
The focus of a parabola is assumed to be S ( a,0 ) and the directrix as the line
| x + a = 0 or x = -a |
To find out the equation of the parabola, we take any point (x,y) on the locus, then we can say that
( x-a )2 +y2 = ( x+a )2
or, ( x-a )2 +y2 = ( x+a )2 - ( x-a )2
thus the standard form of the equation of the parabola can be written as
| y2 = 4ax |
- VERTEX
- AXIS OF THE PARABOLA
- FOCAL CHORD
- DOUBLE ORDINATE
- LATUS RECTUM
- CO-NORMAL POINTS
- DIAMETER OF THE PARABOLA
- PARAMETRIC COORDINATES OF A PARABOLA
The point O (0,0) is referred to as the vertex of the parabola. As we can see from the figure that the tangent to the vertex is the line x = 0. In this case, it is the y-axis.
The line joining the focus S and the vertex is called the axis of parabola. In this case it the x-axis and its equation is therefore y = 0.
Any chord of the parabola that passes through the focus of the parabola is called a focal chord.
Any chord of the parabola that is perpendicular to its axis is called a double ordinate. It may be on any side of the axis.
Latus rectum is that focal chord of the parabola, that is perpendicular to its axis. The length of the latus rectum is given by 4a. Where a is the x coordinate of the focus.
The points on a parabola, the normals at which are concurrent, points are referred to as the co-normal points.
Diameter of a parabola can be described as the line that bisects a system of parallel chords of a parabola.
The parametric coordinates of any point on a parabola are (at2, 2at).
From the equation of the parabola, we can see that the points on a parabola A = (x1,y1) can also be given by A = (y12/4a , y1). If we compare this equation with the parametric form A = (at12 , 2at1), we get y1 = 2at1.
Now the equation of a chord joining the points is given by A = (y12/4a , y1) and B = (y22/4a , y2) is given by
|
y( y1 + y2) - 4ax - y1y2 = 0 |
And the chord joining the points t1 and t2 is given by
| y(t2 - t1) - 2x -2at1t2 = 0 |
FOCAL CHORD
If the chord passes through the focus and hence the point (a,0) then it is the focal chord. Thus, the condition for focal chord is,
| y1y2 = -4a2 |
| Or, t1t2 = -1 |
EQUATIONS OF TANGENTS AND NORMALS
TANGENTS
1. Equation of a tangent at point A(x1,y1)
|
yy1 = 2a( x + x1) |
2. Equation of a tangent in terms of slope is given by,
| y = mx + a/m |
Where m is the slope of the tangent.
Thus the condition for a line to be tangent to a parabola is
| C = a/m |
NORMALS
1. The equation of the normal at point A(x1,y1) is given by
|
y-y1 = -( x - x1) y1/2a |
2. Equation of a normal in terms of slope is given by,
| y = mx -2am - am3 |
Where m is the slope of the tangent.
Thus the condition for a line to be tangent to a parabola is
| C = -2am - am3 |