Parabola

A locus of the points which are at an equal distance from a fixed point and a fixed line is called a parabola. The fixed point is called the focus and the fixed line is called the directrix.
The equation of the parabola with focus at P and directrix ax + by + c = 0 is
√[(x - x1)2 + (y - y1)2] = (ax1 + by1 + c)/[√(a2 + b2)]
  • The straight line joining any two points on the parabola is called the chord of the parabola.
  • The chord of the parabola passing through the focus is called the focal chord of the parabola.
  • The distance of a point on a parabola from the focus is called its focal distance.
  • A straight line drawn perpendicular to the axis of the parabola and terminated at both ends by the parabola is called a double ordinate.
  • The double ordinate passing through the focus of the parabola is called the latus rectum.
Standard forms of parabola
Property\Equation y2 = 4ax y2 = -4ax x2 = 4ay x2 = -4ay
Axis y = 0 y = 0 x = 0 x = 0
Focus (a,0) (-a,0) (0,a) (0,-a)
Directrix x = -a x = a y = -a y = a
Vertex (0,0) (0,0) (0,0) (0,0)
Equation of latus rectum x = a x = -a y = a y = -a

The length of latus rectum in all the above cases is 4a.

Tangent to the parabola y2 = 4ax at P(x1,y1) is
yy1 = 2a(x + x1)
The equation of the normal at this point is
y - y1 = -(y1/2a)(x - x1)
The straight line y = mx + c is tangent to the above parabola if m≠0 and c = a/m.



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