PARABOLA:
Definition:
A Parabola is the set of all points P in the plan that are equidistance from a fixed line and a fixed point in the plan. The fixed point does not lie on the fixed line.
focus:
The fixed is called the directrix of the parabola and the fixed point is called its focus.
Axis of parabola:
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The straight line through the focus and perpendicular to the directrix is called axis of parabola. The point, where the parabola meets its axis, is called the vertex of parabola.
The equation of parabola with focus F(a, 0), a > 0 and vertex at the origin is quite simple. In this case the equation of the directrix ZM is x = -a. If P (x, y) is a point on the parabola, then by definition, P is equidistance from F and ZM. hence |PF| = |PM|i.e.,
(x-a)² + y² = (x + a)²
or
y² = 4ax
which is standard equation of the parabola.
NOW
Let the line through F (a, 0) and perpendicular to the axis of the parabola meet the parabola at L and L' . If y' is ordinate of L, then L (a, y') lies on y² = 4ax.
Therefore,
y'² = 4a²
y' = ±2a
The length LFL '= 4a and is called latusrectum of the parabola.
If a is negative, the opening of the parabola is in the negative direction of the x-axis.
If the focus F (0, a) of a parabola lies on the y-axis, and the vertex is at the origin, then the equation of the parabola is
x² = 4ay
and the curve is symmetric about the y-axis. The orientation of the parabola is upward or downward according to a is positive or negative.
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