Quick revision for IIT-JEE general formulas, Straight lines and Parabola

19 Mar 2009

Here are some shortcuts which you should never ever forget!

1)Area of the triangle made by the line ax+by+c=0 and the co ordinate axes is c²/(2ab)

2)Area of the rhombus formed by ax(+or-)by(+or-)c=0 is 2c²/ab

3)If ABCD is parallelogram

then D=A-B+C

Eg:If (1,6),(2,5),(5,1) are three vertices of a parallelogram

then fourth vertex is (1-2+5,6-5+1) ie(4,2)

4)The foot of the perpendicular (h,k) from (x₁,y₁) to the line ax+by+c=0 is given by

(h-x₁)/a=(k-y₁)/b= -(ax₁+by₁+c)/(a²+b²)

5)The image or reflection of the point (x₁,y₁) in the line ax+by+c=0 is (h,k) where

(h-x₁)/a=(k-y₁)/b= -2(ax₁+by₁+c)/(a²+b²)

(4) and 5) can be applied to 3-D geometry too!)

6)The area bounded by y²=4ax andx²=4by is given by (4a)(4b)/3

7)The area bounded by y²=4ax and the line y=mx is 8a²/(3m³)

8)The area bounded by y²=4ax and its latus rectum x=a is 8a²/3

The most important points of Straight lines
1. The equation of a line parallel to x-axis is of the form y = k.
The equation of a line parallel to y axis is of the form x = k, where k is a constant.

2. If a line makes an angle θ with the positive direction of x-axis and θ ≠π/2, then the slope of the line is given by tan θ.

3. the slope of a line passing through (x1,y1) and (x2,y2) is (y2-y1)/(x2-x1), provided x1≠x2.

4. If two lines have finite slopes m1 and m2
then they are parallel iff m1 = m2
they are perpendicular iff m1*m2 = -1

5. The equation of a line having slope m and y intercept c is y = mx+c

6. The equation of a line having slope m and passing through (x1,y1) is

(y-y1) = m(x-x1)

7. The equation of a line having slope m and passing through (x1,y1)and (x2,y2) is

(y-y1)/(x-x1) = (y1-y2)/(x1-x2)

8.The equation of a line making non-zero intercepts and b on the x and y axes respectively is

(x/a) + (y/b) = 1

9. The equation of a line such that the perpendicular drawn from the origin to the line has length p and inclination α is

x cos α + y sin α = p.

10. The general equation of a line is of the form ax +by +c = o and its slope is –a/b, provided b≠0.

11. If m1 and m2 are the slopes of two lines, then the acute angle θ between them is given by tan θ = |m1-m2|/|1 + m1*m2|, provided m1*m2≠-1.

12. The perpendicular distance of (x1,y1) from the line ax+by+c = 0 is given by |ax1 + by1 + c|/| √(a² + b²)|

13. The point of intersection of two lines, which are not parallel, can be found by solving their equations simultaneously.

14. Family of lines

If u ≡ a1x + b1y +c1 = 0 and
v≡ a2x +b2y +c2 = 0

the u + kv = 0, k ? R represents a family of lines

(i) if u and v are intersecting lines, then u + kv = 0, k ? R represents a family of lines passing through the point of intersection of u =0 and v=0.

(ii) if u and v are two parallel lines, u + kv = 0, k ? R represents a family of straight lines parallel to u =0 and v=0.

Key Points in Parabola

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

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.

TRICK FOR FINDING CUBES OF NUMBERS!!!!!

Cubing numbers that are near a power of ten Such as 13, 104, 1012, etc.

Example: 12³

Step One: Subtract the nearest power of ten from the number: 12 - 10 = 2

Step Two: Double this number and add the number being cubed: (2 x 2) + 12 = 16

Step Three: Subtract from this number the power of ten from step one: 16 - 10 = 6

Step Four: Multiply this number by the answer in step one: 2 x 6 = 12

Step Five: Cube the answer in step one: 2³ = 8

Step Six: Since we're cubing a 2 digit number, add two zeros to the answer in step two: 1600

Step Seven: Add 1 zero to the answer in step four: 120

Step Eight: Add the answers in steps seven and eight to the answer in step five: 1600 + 120 + 8 = 1728

Thus, 12³ = 1728

Cubing numbers that are near a multiple of a power of ten Such as 42, 507, 2008, etc.

Example: 48³

Step One: Keep in mind that the main base is 10 and our working base will be 50

Step Two: Divide the main base by the working base: 10 / 50 = 1/5

Step Three: Subtract the working base from the number being squared: 48 - 50 = -2

Step Four: Double this number and add the number being squared: (-2 x 2) + 48 = 44

Step Five: Divide this number by the square of the ratio found in step two: 44 / (1/5)² = 44 / (1/25) = 44 x 25 = 1100

Step Six: Subtract the working base from the answer in step four: 44 - 50 = -6

Step Seven: Multiply the answer in step three by the answer in step six: -2 x -6 = 12

Step Eight: Divide this number by the ratio in step two: 12 / (1/5) = 12 x 5 = 60

Step Nine: Cube the answer found in step three: -2³ = -8

Step Ten: Since we're cubing a 2 digit number, add two zeros to the answer in step five: 110000

Step Eleven: Add 1 zero to the answer in step eight: 600

Step Twelve: Add the answers in steps ten and eleven to the answer in step nine: 110000 + 600 + -8 = 110592

Thus, 48³ = 110592

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