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$S = y^2 - 4ax \\ S_1 = yy_1 - 2a(x+x_1) \\ S_{11} = {y_1}^2 - 4ax_1 \\ \\ \\ 1.For Parabola \ \Delta \not= 0 \ and \ h^2 = ab .\\ \\ 2. Eccentricity \ = 1. \\ \\ 3.A \ triangle \ with \ vertices \ (x_i , y_i ) \ i=1,2,3 \ lying \ on \ parabola \ S = 0 \ has \ area \ \frac{1}{8a}(y_1-y_2)(y_2-y_3)(y_3-y_1). \\ \\ 4.Area \ of \ triangle \ formed \ by \ pair \ of \ tangents \ and \ chord \ of \ contact \ of \ (x_1,y_1) \ is \ \frac{{({y_1}^2 - 4ax_1)}^{\frac{3}{2}}}{2a}. \\ \\$

5.(a)Condition for line y=mx+c to touch parabola S=0 is c=a/m .

(b)Condition for line y=mx+c to touch the parabola is $c= ma + \frac{a}{m}.$

6.If two tangents with slopes $m_1 \ and \ m_2$ are drawn from to parabola S=0 , then

$m_1+m_2 = \frac{y_1}{x_1} \\ \\ m_1m_2 = \frac{a}{x_1}$ .

7.Point of contact of tangent drawn to parabola is $(\frac{a}{m^2} , \frac{2a}{m}).$

8.(a)Pole of lx+my+n = 0 with respect to parabola S=0 is $(\frac{n}{l} , \frac{-2am}{l}).$

(b)If l=0 then the line is called diameter of parabola.

9.Condition for lines $l_1x+m_1y+n_1=0 \ and \\ \\ l_2x+m_2y+n_2= 0$

to be conjugate with respect to parabola S=0 is $\frac{l_1}{n_1} + \frac{l_2}{n_2} = 2a\frac{m_1m_2}{n_1n_2}$ .

10.Focal distance of a point P on parabola S=0 is

11.Length of chord joining P and Q on parabola S=0 is $(x_1~x_2)\sqrt{1+m^2}$ ,where m is the slope of

the line PQ.

12.Length of chord of contact of with respect to parabola S=0 is $\frac{1}{a}\sqrt{(y_1^2-4ax_1)(y_1^2+4a^2)}.$

$13.At \ a \ distance \ 2a \ from \ vertex \ if \ chords \ are \ drawn \ passing \ through \ point \ K(2a,0) \ then \ \frac{1}{PK^2} + \frac{1}{QK^2} = constant. \\ \\ 14.Slope \ of \ chord \ joining \ P(t_1) \ and Q(t_2) \ is \ \frac{2}{t_1 + t_2}. \\ \\ 15.If \ PQ \ is \ a \ focal \ chord \ then \ t_1t_2=-1. \\ \\ 16.Tangents \ at \ the \ end \ of \ focal \ chord \ intersect \ at \ the \ Directrix. \\ \\ 17.Length \ of \ the \ focal \ chord \ PQ \ is \ a(t+ \frac{1}{t})^2. \\ \\$

18.For a focal chord PQ and focus S , SP , 2a , SQ are in HP i.e. $\frac{1}{SP} + \frac{1}{SQ} = \frac{2}{2a}.$

19.Parametric Coordinates of parabola is

20.Parametric equation of tangent at P(t) is $yt = x + at^2 (slope = \frac{1}{t}).$

21.Point of intersection of tangents at P and Q is

22.Area of triangle formed by any 3 points on parabola is twice the area of triangle formed by tangents at these points.

23.Equation of normal at P(t) is

24.Tangent at one end of focal chord is parallel to normal at the other end.

25.For a normal chord PQ with P and Q

$t_2 = -t_1 - \frac{2}{t_1} .$

26.Three normals can be drawnfrom any point P to the parabola S=0.If are feet of normals on parabola then for co-normal points remember the following table

$t_1 + t_2 +t_3 = 0 \\ \\ t_1t_2+t_2t_3+t_3t_1 = \frac{2a-x_1}{a} \\ \\ t_1t_2t_3 = \frac{y_1}{a} \\ \\ Centroid \ of \ triangle \ formed \ by \ three \ co-normal \ points \ lies \ on \ the \ x-axis.$

27.If normals are drawn from axis of the parabola to it , then one of the co-normal points will be origin (0,0) so $t_3=0 \ t_2 = -t_1 \\ \\ {t_1}^2 = \frac{x_1-2a}{a} \ i.e. \ if \ x_1>2a$

then 2 normals are real and distinct.

28.If normals at P and tersect on the parabola again at R(t) ,then $t_1t_2=2 \ and \ t = -(t_1+t_2).$

29.If chord joining P and subtends a right angle at vertex then

30.If normal chord PQ subtends a right angle at vertex and makes an angle $\theta$ with axis of parabola , then $tan^2\theta = 2.$

31.A circle and a parabola intersect in four points .If , i=1,2,3,4 are points of intersection , then remember the following table

$t_1+t_2+t_3+t_4 = 0 \\ \\ \sum{t_1t_2} = \frac{2g+4a}{a} \\ \\ \sum{t_1t_2t_3} = \frac{-4f}{a} \\ \\ t_1t_2t_3t_4 = \frac{c}{a^2}$

32. If tangent and normal drawn at point P to the parabola S=0 meets the x-axis at T and G respectively , then P,T,G lie on a circle with focus as the center and radius equal to focal distance of P.

33.When a ray of light parallel to the axis of parabola is incident on parabola is reflected through the focus of Parabola.

34.The circle on focal distance as diameter touches the tangent at vertex and intercepts a chord of of length $a\sqrt{1+t^2}$ on normal at point P(t).

35.The locus of feet of perpendiculars from focus on to the tangents of parabola is the tangent at vertex.

36.If the tangent at any point meets the directrix of the parabola at A then angleASP = 90 degree.

37.Parabola can be expressed as $(distance \ of \ point \ from \ axis)^2 = (Length \ of \ Latus Rectum)(distance \ of \ same \ point \ from \ tangent \ at \ the \ vertex).$

38.Equation of normal to the parabola with a given slope m is

39.Two parabolas are said to be equal if they have the same latus rectum.

40.Length of a focal chord making an angle $\alpha$ with the axis of parabola is $4acosec^2\alpha.$

41.Length of subnormal is constant for all the points on the parabola and is equal to the semi latus rectum.

42.If the tangents at P and Q meet in T , then TP and TQ subtend equal angles at focus S and which implies triangles SPT and STQ are similar.

43.Area enclosed between the Parabolas $y^2=4ax \ and \ x^2=4by \ is \frac{16ab}{3}.$

44.Area enclosed between a line y=mx+c and parabola S=0 is $\frac{8}{3} \ \frac{a^2}{m^3}.$

45.Conveerse to the point no 33 any line passing through the focus of the parabola emerges parallel to the axis of the parabola after reflection at the curved surface.This property is used in many places .For example head lights of the car will have the parabolic structure with bulb at the focus of the parabola.

Phewwww.......friends this is my first Latex embedded presentation and my first self-written article.I mean there may be some errors even though I have checked it twice.So please if u find them , bring them to my notice.

Hope I've covered all the important points.And friends this article is made assuming that you know the definitions and basics of parabola.

And dun forget to comment .

~INDIAN

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