a parabola is formed by the intersection of a plane with a cone when the cone intersects parallel to...
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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 ParabolaA 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
axyax 22)(222 )()( axyax
22222 22 aaxxyaaxx
axy 42 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.
y2 = 4ax‘a’ is positive
‘a’ is negative
In other form
‘a’ is negative
‘a’ is positive
x2 = 4ay
CHORD SECANTTANGENT
Equation of the tangent at the point P(x1,y1) to the parabola y2=4ax
is given by ;
yy1=2a(x+x1)
P(x1,y1)
Slope of tangent = 2a/y1
Equation of tangent
Equation of the normal at the point P(x1,y1) to the parabola y2=4ax
is given by ;
y - y1 = y1 / 2a ( x - x1 )
P(x1,y1)
NORMAL
Slope of normal = -y1/ 2a
Equation of normal
Equation of tangent and normal in parametric form
Equation of the tangent to y2 = 4ax at the point (at2, 2at) is given by:
yt = x+at2
Equation of the normal to y2 = 4ax at the point (at2, 2at) is given by: y = –tx + 2at + at3.
Equation of normal in slope form
Equation of the normal to y2 = 4ax in slope form is given by:
y = mx - 2am - am3,
where m is the parameter and (am2, -2am) is the point of contact. This cubic in m has three roots say; m1, m2, m3, 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, m1+ m2+ m3 = 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(x1,y1) to the parabola y2=4ax is given by:
(y2-4ax)(y12-4ax1) = [yy1-2a(x+x1)]2
PAIR OF TANGENTSCHORD OF CONTACT
P(x1,y1)
and equation of chord of contact is given by:
yy1=2a(x+x1)
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 y2=4ax whose mid point is P(x1,y1) is given by ;
yy1-( x + x1 ) = y12 – 4ax1
P(x1,y1)
Parabolas show up in the architecture of bridges, in fountains etc
