PARABOLA

Definition : 

 A parabolla is the locus of the point in plane equidistant from a fixed point and a fixed line in that plane . The fixed point is called the focus and the fixed straight line is called the directrix.

Standard Equation Of The Parabola :

Equation of the parabola in the standard form y² = 4ax.

Let S be the focus and d be the directrix of the parabola . Let SZ be perpendicular to the directrix . Bisect SZ at the point O. By the definition of parabola the midpoint O is on the parabola. Take O as the origine , line OS as the X- axis and and the line through O perpendicular to OS as the Y-axis .       

Let SZ = 2a , a > 0 .

Then the coordinates of the focus S are (a, 0) and the coordinates of Z are (-a, 0).

The equation of the directrix d is 

x = -a , i.e. x + a = 0

Let P(x, y) be any point on the parabolla . Draw segment PM perpandicular to the directrix d .

∴ M = (-a, y) 

By using distance formula we have

By focus - directrix property of the parabolla SP = PM

Squaring both side (x - a)²  + y² = (x - a)² 

That is  x² - 2ax +a² +  y² = x²  + 2ax + a² 

That is y² = 4ax  (a > 0)

This is the equation of parabolla in standard form.

Tracig Of The Parabola   y² = 4ax  (a > 0) 

1) Symmetry : Equation of the parabola can be writen as y = 士 2√ax  that is for every value of a , there are two values of y which are negative of each other . Hence parabola is symmmetric about X-axis.

2) Region  :  For every x < 0, the value of y is imaginary therefore entire part of the curve lies to the right of  Y=axis.

3) Intersection With the Axis : For x = 0 we have y = 0 , therefore the curve meets the coordinates axes at the origin O(0, 0).

4) Shape of Parabola : As x→∞ , y→∞. Therefore the curve extends to infinity as x grows large and open in the right half plane shape of the parabola y² = 4ax  (a > 0) . 

Some other standard forms of parabola

Parameter :

    If the co-ordinates of point on the curve are expressed as function of a variable, that variable is called the parameter for the curve.

Parametric expression of standard parabola y² = 4ax :

   x =at² , y = 2at are the expression which satisfies given equation   y² = 4ax for any real value of t that is  y² =  (at)² = 4  a² t² = 4a (at)² = 4ax where t is a parameter .

General forms of the equation of a parabola :

      If the vertex is shifted to the point (h, k) we get the following form 

  (y - k)² = 4a (x- h) , this represents a parabola whose axis of symmetry is y - k = 0 which  is parallel to the X- axis , vertex is at (h, k) and focus is at (h + a, k) and directrix is x  = h - a . It can be reduced to the form x = A y² + By + C  OR   Y² = 4a X , where X = x- h , Y = y - k .

Tangent : 

    A straight line which intersects the parabola in coinsident point is called a tangent of the parabola.                            

Point Q moves along the curve to the point P. The limiting position of secant PQ is the tangent at P.

A tangent to the curve is the limiting position of a secant intersecting the curve in two points and moving so that those points of intersection come closer and finally coincide . 

Tangent at a point on a parabola :

      Let us find the equation of tangent to the parabola at a point on it in cartesion form and in parametrics form. 

  We find the equation of tangent to the parabola  y² = 4ax at the point P (x_{1 ,} y₁) on it . Hence, obtain the equation of tangent at P(t).

Equation of the tangent to the curve y = f(x) at point (x_{1 ,} y₁) on it is .

y - y = [f ' (x)]_{(x1 , y1)}    (x - x₁) [f ' (x)] _{(x1 , y1)} 

We need to the slope of the tangent at P(x_{1 ,} y₁) . From theory of derivatives of a function, the slope of the tangent is dy/dx  at ( x_{1 ,} y₁).

Condition of Tangency :

      To find the condition that the line y = mx is tangent to the parabola y² = 4ax . Also to find the point of contact.

Equation of the line is y = mx + c

∴  mx - y + c = 0 .....(I)

equation of the tangent at P ( x_{1 ,} y₁) to the parabola  y² = 4ax  is yy₁= 2a (x + x₁) 

∴  2 a x - y₁ + 2 a x₁ = 0 ......(II)

If the line given by equation(I) is a tangent to the parabola at ( x_{1 ,} y₁) . Equation (I) and eqaution (II) represent the same line .

 Compairing the co-efficients of like terms in equations (I) and (II) 

we get 2a/m = - y₁ /-1 = 2a x₁ / c

∴  x₁  = c/m    and  y₁  = 2a/m

But the point P ( x_{1 ,} y₁) lies on the parabola 

∴    y₁² = 4a  x₁

∴    (2a/m)²   = 4a (c/m)

   4a² /m²   =  4a(c/m)

∴    c = a/m 

This is required condition of tangency .

Tangent from a point to a parabola : 

     In general two tangent can be drawn to a parabola  y² = 4ax  from any point in its plane.

Let P ( x_{1 ,} y₁)  be any point in the of parabola . Equation of tangent to the parabola y² = 4ax  is 

y = mx + a/m

Sience the tangent passes through P ( x_{1 ,} y₁) , we have  y₁ = mx₁ + a/m

∴  my₁ = m² x₁ +a 

m² x₁ - my₁ + a = 0..........(I)

x₁m²  - y₁m +a = 0

This is quadratic equation in m and in general it has two roots , m₁ and m₂ which are the slopes of two tangents.

Thus, in general , two tangent can been drwan to a parabola from a given point in its plane.

If the tangent drawn from P are mutually perpandicular we have 

m₁m₂  = - 1

from equation (I) m₁m₂ = a/x .....(product of roots)

∴  a / x₁  = -1

∴   x₁= -a 

which is the equation of directrix.

Thus, the locus of the point, the tangent from which to the parabola are perpandicular to each other is the directrix of the parabola.