Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Elementary Concepts of Parabola

Definition:

A parabola is the locus of a point, whose distance from a fixed point (focus) is equal to perpendicular distance from a fixed straight line (called directrix).

Some Standard Forms of Parabola

For  y^2 =4ax

(a) Vertex is (0 , 0)

(b) Focus is (a , 0)

(c) Axis is y =0

(d) Directrix is  x+a=0

Focal Distance:

The distance of a point on the parabola from the focus.

Focal Chord:

A chord of the parabola, which passes through the focus.

Double Ordinate:

A chord of the parabola, which is perpendicular to the axis of the symmetry.

Latus Rectum:

A double ordinate passing through the focus or a focal chord perpendicular to the axis of the parabola is called the latus rectum. (L.R)

For  y^2 =4ax

 \implies length of L.R  =4a

 \implies Ends of L.R are L (a ,2a) and  L' (a , -2a)

Note:

(i) Perpendicular distance from the focus on the directrix = half the latus rectum.

(ii) Vertex is the middle point of the focus and the point of intersection of directrix and axis.

(iii) Two parabolas are said to be equal if they have the same Latus Rectum.

Key Point:

\to   The equation of a parabola with directrix  x=-a and focus (a , 0) is y=4ax .

\to The parametric equation of a parabola with directrix x=-a and focus ( a ,0) is  x=at^2, \quad y=2at.

Example:

Find the equation of the parabola whose focus is at  ( -1, -2) and the directrix is  x-2y+3=0.

Answer:

Let  P (x,y) be any point on the parabola whose focus is  F (-1, -2)   and the  directrix  x-2y+3=0  

PM =PF

 \implies PM^2 =PF^2

\implies \left( \dfrac{x-2y+3}{\sqrt{1+4}} \right)^2= (x+1)^2+(y+2)^2

 \implies (x-2y+3)^2=5[x^2+y^2+2x+4y+5]

 \implies x^2+4y^2+9-4xy+6x-12y=5x^2+5y^2+10x+20y+25

\implies 4x^2+y^2+4xy+4x+32y+16=0

\therefore This is the equation of parabola.

Parametric form of Parabola:

(i) For  y^2 =4ax, \quad x=at^2, \quad y=2at

(ii) For  y^2 =-4ax, \quad x=-at^2, \quad y=2at

(iii) For x^2 =4ay, \quad x=2at, \quad y=at^2

(iv)  For x^2 =-4ay, \quad x=2at, \quad y=-at^2

Example:

Find the parametric equation of the parabola  (x+1)^2=-6(y+2)

Solution:

Given  (x+1)^2=-6(y+2)

 \therefore 4a=-6 \implies a = -\dfrac{3}{2}

 at^2=y+2 (i)

 2at=x+1 (ii)

From (i) y=-2+at^2=-2-\dfrac{3}{2}t^2

From (ii)  x=2at-1=2\times -\dfrac{3}{2}\cdot t-1

=-1-3t

 \therefore x=-1-3t, \quad y=-2-\dfrac{3}{2}t^2

Position of a Point Relative to a Parabola:

The position of a point defined by the point  P (x_1, y_1) lies outside, inside or on the parabola  y^2=4ax according as the expression   y_1^2-4ax_1 is positive, negative or zero.

Example:

Check the point  ( 4, 5) situated to the parabola  y^2=4x

Solution:

 y^2 -4x=0

F_1 : y_1^2 -4x_1=25-16=9 > 0

 \therefore (4, 5) lies outside the parabola  y=4x

Line and a Parabola:

The line  y=mx+c meets the parabola y^2=4ax in two points real, coincident or imaginary according as a  a > \operatorname{cm}, \quad a =\operatorname{cm}\quad a< \operatorname{cm}  respectively.

Length of the chord intercepted by the parabola on the line y=mx+c is :

 \dfrac{4}{m^2}\sqrt{a(1+m^2)(a-mc)}

Note:

 \to The equation of a chord joining t_1 and  t_2 is  2x-(t_1+t_2)y+2at_1t_2=0

\to If t_1 and  t_2 are the ends of a focal chord of the parabola y^2=4ax then t_1t_2=-1 . Hence the co-ordinates at the extremities of a focal chord can be taken as  (at^2, \quad 2at)  and  \left( \dfrac{a}{t^2}, -\dfrac{2a}{t} \right)

\to Length of the focal chord making an angle  \alpha with the x -axis is  4a \csc^2 \alpha

AB=4a\csc^2 \alpha

Tangent to the Parabola  y^2=4ax

An equation of the tangent at a point parabola can be obtained by the replacement method or derivatives.

In the replacement method, the following changes are x^2 \longrightarrow xx_1, \quad y^2\longrightarrow yy_1, \quad 2xy \longrightarrow xy_1+x_1y, \quad 2x\longrightarrow x+x_1, \quad 2y \longrightarrow y+y_1

So, the tangents are:

(i)  yy_1=2a(x+x_1)   at the point  (x_1,y_1)

(ii)  y=mx+ \dfrac{a}{m} (m \neq 0)  at \left( \dfrac{a}{m^2}, \dfrac{2a}{m} \right)

(iii) ty=x+at^2   at  (at^2, 2at)

(iv) Point of intersection of the tangents at the point t_1 and t_2 is \{at_1t_2, a(t_1+t_2)\}

Normal to Parabola:

Normal is obtained using the slope of the tangent.

 

Slope of tangent at  (x_1, y_1)= \dfrac{2a}{y_1}

 \implies Slope of normal =- \dfrac{y_1}{2a}

(i)y-y_1=-\dfrac{y_1}{2a}(x-x_1) at (x_1 , y_1)

(ii)  y=mx-2am-am^3 at (am^2, -2am)   

(iii)  y+tx=2at+at^3 at (at^2, 2at)

Note:

 \to Point of intersection of normals at t_1   and  t_2  is \left( a(t+t+t_1t_2),-at_1t_2(t_1+t_2) \right)

\to If the normals to the parabola  y^2=4ax at the point  t_1 , meets the parabola again at the point t_2, then  t_2 =-\left( t_1+\dfrac{2}{t_1} \right)

 \to If the normals to the parabola y^2=4ax at the points t_1 and  t_2 intersect again on the parabola at the point  t_3, then  t_1t_2=2 ; t_3 =-(t_1+t_2) and the line joining t_1   and t_2 passes through a fixed point (-2a, 0)

Application of Parabola:

\to A satellite dish receiver is the shape of a Parabola.

 \to An arched underpass of a road has the shape of a Parabola.

\to A soup bowl has a cross-section with a parabolic shape.

Some Important Points:

(i) The tangent and the normal at a point on the parabola are the bisectors of the angle between the focal radius and the perpendicular from that fixed point on the directrix.

From this, we conclude that all the rays emitting from the focus will remain parallel to the axis of the parabola after reflection.

(ii) The portion of the tangent to a parabola cut off between the directrix and the curve subtends a right angle at the focus.

(iii) Any tangent to a parabola and the perpendicular on it from the focus meet on the tangent at the vertex.

(iv) Semi Latus Rectum of the parabola  y^2=4ax , is the harmonic mean between segments of any focal chord of parabola.

(v) The area of the triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents at these points.

(vi) Length of subnormal is constant for all points on the parabola and is equal to the semi-latus rectum.

Comparative Study of Four Parabolas:

  Parabola y^2=4ax y^2=-4ax x^2=4by x^2=-4by
1 Focus (a ,0 ) (-a ,0) (0 , b) (0, -b)
2 Directrix x+a=0 x-a=0 y+b=0 y-b=0
3 Axis x -axis  x-axis  y-axis y -axis
4 Vertex Origin Origin Origin Origin
5 Latus Rectum 4a 4a 4b  4b
6 Ends of Latus Rectum (a, 2a), (a , -2a)  (-a, 2a), (-a ,-2a)  (2b, b) (-2b, b)  (2b ,- b), (-2b, -b)
7 Lies in I and IV quadrants I and III quadrants I and II quadrants III and IV quadrants

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