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PARABOLA

Synopsis 

Definition: The locus of a point which moves in a plane such that the ratio of its distance from a fixed point to its perpendicular distance from a fixed straight line not passing through given fixed point is always constant is called conic section or conic.
Here the fixed point is called focus and usually it is denoted by 'S'. Fixed line is called directrix and fixed ratio is called eccentricity and it is denoted be 'e'.
(i) If e = 1, the conic is called Parabola.
(ii) If e < 1, the conic is called Ellipse.
(iii) If e > 1, the conic is called Hyperbola.

 

Equation of conic section
Suppose lx + my + n = 0 is directrix, S(α, β) is focus, e is eccentricity of conic.

    
        SP = ePM
        SP2 = e2 PM2

       If we simplify the above equation that becomes of the form
       ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
 ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 ...............(1) represents conic equation
      ∆ = abc + 2fgh - af− bg2 − ch2 ≠ 0
     (i) If  h2 = ab; (1) represents Parabola.
     (ii) If  h2 < ab; (1) represents Ellipse.
     (iii) If  h2 > ab; (1) represents Hyperbola.

 

STANDARD EQUATION OF PARABOLA 

Let S is a focus and L = 0 is directrix.

Draw the line passing through S, perpendicular to the directrix to meet the directrix at Z.

Let A be the mid point of SZ.
clearly SA = AZ
 A lies on the Parabola.
Take   as X−axis, and the line passing through A, perpendicular to  as Y−axis.

Let S is (a, 0) then equation of directrix is x + a = 0.
Let P(x, y) be any point on the Parabola
then SP = PM
SP2 = PM2
(x − a)2 + y2 = (x + a)2
y2 = (x + a)2 − (x − a)2


          
Some terms related to Parabola:
Axis:
The straight line passing through focus perpendicular to directrix is called Axis.
Vertex: The point of intersection of axis and Parabola is called 'Vertex'.
     Here vertex is (0, 0)
Focal chord: Any chord passing through focus is called focal chord.
Double ordinate: Suppose P is any point on Parabola. The foot of perpendicular from point P to axis is called ordinate, if produced to meet the curve again at P', then PP' is called double ordinate.
Latus rectum: A double ordinate passing through focus is called Latus rectum.
     For Parabola Latus rectum = 4a

Focal distance: If P(x, y) is any point on the Parabola then the distance from focus 'S' to point is called focal distance.
S(a, 0) is focus, then focal distance = x + a
Equation of latus rectum is x = a
Extremities of latus rectum are (a, 2a), (a, −2a)
Parametric form: For any value of t  R
x = at2, y = 2at satisfies the equation y2 = 4ax.
...  (at2, 2at) are parametric coordinates.

    Notation: S = y2 − 4ax, P(x1, y1), Q(x2, y2)
    S1 = yy1 − 2a(x + x1)
    S12 = y1y2 − 2a (x1 + x2)
    S11 = y12 − 4ax1
P(x1, y1), Q (x2, y2) are two points on the Parabola y2 = 4ax then equation of chord
passing through P, Q is S1 + S2 = S12
       yy1 − 2a (x + x1) + yy2 − 2a(x + x2) = y1y2 − 2a (x1 + x2)
P(at12, 2at1), Q(at22, 2at2) then equation of chord passing through PQ is
    y(t1 + t2) = 2x + 2at1t2
    If the chord passing through focus S(a, 0), then
     0(t1 + t2) = 2a + 2at1t2
     2a(t1t2 + 1) = 0


      
  
...  P(t1)Q(t2) is focal chord, then

  

     
P(t1)Q(t2) is focal chord, then SP, 2a, SQ are in Harmonic Progression.
Length of Focal chord drawn at P(t) is 
      
...    Minimum length of focal chord is latus rectum.
Equation of tangent drawn to Parabola, S = y2 − 4ax = 0 at P(x1, y1) is S1 = 0
yy1 − 2a(x + x1) = 0


 

Equation of tangent to the Parabola with slope m is, y = mx + 
 The condition that y = mx + c is tangent to the Parabola y2 = 4ax is c = 
Equation of tangent to the Parabola drawn at P(t) is yt = x + at2
Tangents drawn at P(t1), Q(t2) to Parabola intersect at T(a(t1t2), a(t1 + t2))
Line joining midpoint of PQ, T is parallel to axis
∆SPT, ∆STQ are similar triangles


Locus of point of intersection of perpendicular tangents to Parabola is directrix.
P(x1, y1) lies outside, on or inside the Parabola y2 = 4ax, according as y12 − 4ax1 > = < 0

If P(x1 , y1) lies outside the Parabola then there exist two tangents from P(x1, y1) to y2 − 4ax = 0 and pair of tangents are S12 = SS11.
If P(x1, y1) lies on directrix then the pair of lines are perpendicular.
Area of the triangle formed by tangents at P, Q, R is half of the area formed by P, Q, R as vertices.
Circumcircle of the triangle formed by tangents at P, Q, R passes through focus, and its orthocentre lies on directrix.
Tangent at P(t1), Q(t2) to Parabola intersects on the line parallel to axis passing through mid point of PQ.
The image of focus with respect to tangent at any point of Parabola lies on directrix.
Foot of the perpendicular drawn from focus on to any tangent lies on directrix.
Tangent at P(t1) to Parabola intersects the axis at 'T' then mid point of PT lies on directrix.

 

Mid point of PT = Mid point of SQ = R
 The segment of tangent between the curve and directrix subtend right angle of focus.


     
 Tangents drawn at extremities of focal chord intersects on directrix and tangents are perpendicular.
 If the chord joining P(t1), Q(t2) subtend right angle at vertex then t1.t2 = − 4
 If P(t1), Q(t2) is focal chord then length at the focal chord is equal to two times distance from mid point of PQ to directrix.
 If the tangents at Q, R intersects at P, then the chord QR is called chord of contact of P with respect to Parabola.

 

If QR is focal chord, then P lies on directrix and tangents at R, Q are perpendicular.
If P(x1, y1) is a point outside the Parabola, chord of contact of P(x1, y1) with respect to Parabola is S1 = 0
Tangents are drawn from the point P(x1, y1) to the Parabola y2 = 4ax, then the length of chord of contact is, 


     
  Area of the triangle formed by the tangents drawn from (x1, y1) to y2 = 4ax (a > 0) and chord of contact is   
If θ is angle between tangents drawn at an external point P(x1, y1) to Parabola y2 = 4ax, then tan θ =     if x1 + a = 0 
     Perpendicular tangents intersect on directrix.
Equation of chord with mid point P(x1, y1) to Parabola is S1 = S11.

y = mx + c is tangent to y2 = 4ax, then  c =   and point of contact is 

 


 y = mx − 2am − am3 is normal to the Parabola y2 = 4ax at (am2, −2am), m  R
 y + xt = 2at + at3 is normal to y2 = 4ax at (at2, 2at)
 Normal draw to y2 = 4ax at P(t1) again meet the Parabola at Q(t2), then 
     
 If the Normal Chord at P(t) to y2 = 4ax subtends right angle at vertex, then t2 = 2


 Tangent and normal at P(t) drawn to y2 = 4ax is internal and external angular bisectors of  .
 If the tangent and normal at P(t) meets the axis at T, G then SP = ST = SG.
 If N is foot of the perpendicular from P to axis then NG = 2a is constant and is called length of subnormal.
 All rays of light coming from the positive direction of X-axis and parallel to the axis of the Parabola after reflection passes through the focus of the Parabola.
 Point of intersection of normals at P(t1), Q(t2) to y2 = 4ax is, 
     

 If y = mx + c is normal to y2 = 4ax then c = −2am − am3

 

y = mx − 2am − am3 is normal to y2 = 4ax
If y = mx − 2am − am3 passing through P(h, k), then am3 + m(2a − h) + k = 0 ............... (1)
Equation (1) is cubic in m, it has three roots say m1, m2, m3
m1 + m2 + m3 = 0,
m1m2 + m1m3 + m2m=   
m1m2m3
Corresponding to each of these three roots, we have one normal which is passing through P(h, k).
...  From any point P(h, k), we can draw maximum three normals to Parabola y2 = 4ax.
If the normals at A, B, C intersects at P(h, k), then A, B, C are called conormal points.
Circumcircle of conormal points passes through vertex of Parabola.
The algebraic sum of the slopes of three concurrent normals is zero.

 

Locus of midpoints of parallel chords is called diameter of the Parabola.
If m is slope of parallel chords then equation of diametre is y = 
Diameter of Parabola is parallel to axis.

P(x1, y1) is point and y2 = 4ax is Parabola, locus of point of intersection tangents drawn at extremities of the chords passing through the point P(x1, y1) is straight line is called polar of P(x1, y1) with respect Parabola and point is called Pole of polar.
Polar of P(x1, y1) with respect to S = y2 − 4ax = 0 is S1 = 0.
Two points P(x1, y1), Q(x2, y2) are said to be conjugate points with respect Parabola y2 = 4ax if polar of one point passing through another one.
If P(x1, y1), Q(x2, y2) are conjugate points with respect S = y2 − 4ax then S12 = 0.
Two lines are said to be conjugate lines with respect to Parabola if pole of one line lies on another one. 

Posted Date : 19-02-2021

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