A parabola is formed by the intersection of a plane with a cone when the cone intersects parallel to the slant height of the cone.

On a cartesian plane, the set of points that describe a parabola is defined using a point called the FOCUS and a line called the DIRECTRIX. DIRECTRIX FOCUS The distance of a given point on the parabola from the focus is equal to the distance of that same point to the directrix. When that point is the vertex that distance has a special significance. It defines an important parameter for the parabola known as ‘a’. The distance from the focus to the vertex or from the directrix to the vertex is ‘a’. This value plays a role in defining the equation of the parabola. VERTEX

Definition of Parabola A parabola is the locus of a variable point on a plane so that its distance from a fixed point (the focus) is equal to its distance from a fixed line (the directrix x = - a). focus F(a,0) P(x,y) M(-a,0)x y O N(-a,y)

From the definition of parabola, PF = PN standard equation of a parabola ‘a’ is positive

The equation for a parabola with a vertex at the origin can have one of two formats depending on whether it opens vertically or horizontally. y 2 = 4ax ‘a’ is positive‘a’ is negative

In other form ‘a’ is negative ‘a’ is positive x 2 = 4ay

CHORD SECANT TANGENT Equation of the tangent at the point P(x 1,y 1 ) to the parabola y 2 =4ax is given by ; yy 1 =2a(x+x 1 ) P(x 1,y 1 ) Slope of tangent = 2a/y 1 Equation of tangent

Equation of the normal at the point P(x 1,y 1 ) to the parabola y 2 =4ax is given by ; y - y 1 = y 1 / 2a ( x - x 1 ) P(x 1,y 1 ) NORMAL Slope of normal = -y 1 / 2a Equation of normal

Equation of tangent and normal in parametric form Equation of the tangent to y 2 = 4ax at the point (at 2, 2at) is given by: yt = x+at 2 Equation of the normal to y 2 = 4ax at the point (at 2, 2at) is given by: y = –tx + 2at + at 3.

Equation of normal in slope form Equation of the normal to y 2 = 4ax in slope form is given by: y = mx - 2am - am 3, where m is the parameter and (am 2, -2am) is the point of contact. This cubic in m has three roots say; m 1, m 2, m 3, which shows that three normals can be drawn from any point to a parabola of which at least one must be real for imaginary roots of an equation with real coefficients occur in conjugate pairs. Also, m 1 + m 2 + m 3 = 0 i.e. the sum of the ordinates of the feet of the normals from a given point is zero.

Combined equation of the pair of the tangents at the point P(x 1,y 1 ) to the parabola y 2 =4ax is given by: (y 2 -4ax)(y ax 1 ) = [yy 1 -2a(x+x 1 )] 2 PAIR OF TANGENTS CHORD OF CONTACT P(x 1,y 1 ) and equation of chord of contact is given by: yy 1 =2a(x+x 1 ) Equation of pair of tangents and chord of contact

Equation of chord in terms of its mid point. Equation of the chord of the parabola y 2 =4ax whose mid point is P(x 1,y 1 ) is given by ; yy 1 -( x + x 1 ) = y 1 2 – 4ax 1 P(x 1,y 1 )

Parabolas show up in the architecture of bridges, in fountains etc