1 Standard Parabola :
| Imp.Terms | y2=4ax | y2 =- 4ax | x2 = - 4ay | x2 =-4ay |
| Vertex (v) | (0,0) | (0,0) | (0,0) | (0,0) |
| Focus(f) | (a,0) | (-a,0) | (0,a) | 0,-a) |
| Directrix (D) | x=-a | x=a | y=-a | y=a |
| Axis | y=0 | y=0 | x=0 | x=0 |
| L.R. | 4a | 4a | 4a | 4a |
| Focal distance | x + a | a-x | y+a | a-y |
Parametric
coordinates | (at2, 2at) | (-at2, 2at) | 2at, at2 | (2at , -at2) |
| Parametric | x=at2 | x=-at2 | x = 2at | x = 2at |
| Equations | y=2at | y=2at | y = 2at2 | y=-at2 |
2 Special Form of Parabola :
* Parabola which has vertex at (h,k), latus rectum e and axis parallel to x-axis is
(y-k)2 = e(x-h)
Þ axis is y = k and focus at {h+(e/4),k}
* Parabola which has vertex at (h,k), latus rectum e and axis parallel to y-axis is (x-h)2 = e(y-k)
Þ axis is x = h and focus at { h,k + (e/4) }
* Equation of the ax2 + bx + c = y represents parabola.
 
And axes parallel to y-axis
Note :
parametric equation of parabola (y-k)2 = 4a (x-h) are x=h+at2 , y = k + 2at
3 Position of a point:Position of a point (x1, y1) and a line w.r.t. parabola y2 = 4ax.
* The point (x1, y1) lies outside, on or inside the parabola y2 = 4ax according as y21 - 4ax1 > , = or < 0
* The line y = mx + c does not intersect, touches, intersect a parabola y2 = 4ax according as c > = < a/m
Note:
Condition of tangency for parabola y2 = 4ax, we have c = a/m and for other parabolas check disc. D = 0
4 Equation of tangent in different forms :
(i) Point Form / Parametric form
Equation of tangent of all other standaerd parabolas at (x1, y1 ) / at t(parameter)
| Equation of parabola | tangent at (x1, y1) | Parametric Coordinates’t’ | Tangent of ‘t’ |
| y2 =4ax | yy1 = 2a (x+x1) | (at2, 2at) | ty = x +x at2 |
| y2 = -4ax | yy1 = - 2a(x+x1) | (-at2, 2at) | (tr =- x + at2 |
| x2 = 4ay | xx1 = 2a (y+y1) | (2at, at2) | tx = y + at2 |
| x2 = -4ay | xx1 =- 2a(y+y1) | (2at,-at2) | tx =- y + at2 |
(ii) Slope form:
Equations of tangent of all other parabolas in slope form
Equation of
parabolas | point of contact
in terms of slope (m) | Equations of tangent
in terms of slope (m) | Condition of
Tangency |
| y2 = 4ax | {(a/m2) , (2a/m)} | y= mx + (a/m) | c= (a/m) |
| y2 = - 4ax | {-(a/m2), (2a/m)} | y=mx – (a/m) | c=- (a/m) |
| x2 = 4ay | (2am,am2) | y=mx – am2 | c=-am2 |
| x2 =- 4ay | (-2am, - am2) | y = mx + am2 | c= am2 |
5 Point of intersection of tangent: Point of intersection of tangent at any two points P(at21, 2at1) and Q(at22, 2at2) on the parabola y2 = 4ax is (at1t2, a (t1 + t2)) i.e. (a(G.M.)2, a(2A.M.))
6 Combined equation of the pair of tangents:Combined equation of the pair of tangents drawn form a point to a parabola is SS’ = T2 , where S = y2 - 4ax, S’ = Y21 - 4ax1 and T = yy1 - 2a(x+x1)
7 Equation of normal in different forms
(i) Point form / Parametric form
Equation of normal of all other standard parabolas at (x1, y1) / at t(parameter)
| Eqn. Of Parabola | Normal at (x1, y1) | point ‘t’ | Normals at ‘t’ |
| y2 = 4ax | y-y1 = (-y1 / 2a) (x – x1 | (at2, 2at) | y+tx = 2at + at3 |
| y2 = -4ax | y-y1 = (y1/2a) (x-x1) | (-at2, 2at) | y-tx = 2at+ at3 |
| x2 = 4ay | y-y1 = - (2a/x1)(x-x1) (2at, at2) | x+ty = 2at+ at3 |
| x2 = -4ay | y – y1 = (2a/x1) (x-x1) | x-ty = 2at + at3 |
(ii) Slope form Equation of normal, point of contact , and condition of normality in term of slope (m)
| Eqn. Of Parabola | point of contact | Equations of normal | Condition of Normality |
| y2 = 4ax | (am2, -2am) | y= mx-2am – am3 | c= 2am – am3 |
| y2 =-4ax | (-am2, 2am) | y= mx+ 2am + am3 | c = am + am3 |
| x2 = 4ay | {-(2a/m),(a/m2)} | y= mx + 2a + (a/m2) | c = 2a + (a/m2 |
| x2=-4ay | {(2a/m), -(a/m2)} | y = mx -2a – (a/m2) | c = - 2a – (a/m2) |
Note:
(i) In circle normal is radius itself .
(ii) Sum of ordinates (y coordinate) of foot of normals though a point is zero.
(iii) The centroid of the triangle formed by taking the foot of normals as a vertices of concurrent normals of y2 = 4ax lies on x-axis .
8 Condition for three normals form a point:Condition for three normals form a point (h,0) on x-axis to parabola y2 = 4ax
(i) we get 3 normals if h > 2a
(ii) we get one normal if h £ 2a
(iii) If point lies on x-axis, then one normal will be x-axis itself.
9 (i) If normal of y2 = 4ax at t1 meet the parabola again at t2 = - t1 - (2/t1)
(ii) The normals to y2 = 4ax at t1 and t2 intersect each other at the same parabola at t3, then t1t2 = 2 and t3 = - t1 - t2
10 (i) Equation of focal chord of parabola y2 = 4ax at t1 is
If focal chord of y2 = 4ax cut (intersect) at t1 and t2 then t1 t2 = - 1 (t1 must not be zero)
(ii) Angle formed by focal chord at vertex of parabola is
tan q = (2/3) |t2 - t1|
(iii) Intersecting point of normals at t1 and on the parabola y2 = 4ax is (2a + a(t21 + t22 + t1t2) , - at1t2 (t1 + t2))
11 Equation of chord of parabola:Equation of chord of parabola y2 = 4ax which is bisected at (x1, y1) is given by T = S1
12 The locus of the mid point of a system:The locus of the mid point of a system of parallel chords of a parabola is called its diameter.its equation is y = (2a/m)
13 Equation of polar at the point:Equation of polar at the point (x1, y1) with respect to parabola y2 = 4ax is same as chord of contact and is given by
T = 0 i.e. yy1 = 2a(x + x1)
Coordinates of pole of the line e x my + n =0 w.r.t. the parabola y 2 = 4ax is {(n/e), (-2am / e)}
14 Diameter: It is locus of mid point of parallel chords and equation is given is by T = S1
15 Important results for Tangent:
(i) Angle made by focal radius of a point will be twice the angle made by tangent of the point with axis of parabola
(ii) The locus of foot of perpendicular drop form focus to any tangent will be tangent at vertex.
(iii) If tangent drawn at ends point of a focal chord are mutually perpendicular then their point of intersection will lie on directrix.
(iv) Any light ray travelling parallel to axis of the parabola will pass though focus after reflection through parabola
(v) Angle included between focal radius of a point and perpendicular form a point to directx will be bisected of tangent at that point also the external angle will be bisected by normal.
(vi) Intercepted portion of a tangent between the point of tangent and directrix will make right angle at focus.
(vii) Circle drawn on any focus radius as diameter will touch tangent at vertex.
(viii) Circle drawn on any focal chord as diameter will touch directrix.
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