(vi) cos2A + cos2B + cos2C = 1 – 2cosA cosB cosC (vii) tanA + tanB + tanC = tanA tanB tanC (viii) cotB cotC + cotC cotA + cotA cotB = 1 (ix) tan A/2 tan B/2 = 1
(x) cot A cot B =1
(xi) cot A/2 = cot A/2 11.
Some useful series:
(i)
n 1 n sin 2 2 , 2 n
sin
sin + sin ( + ) + sin( + 2) + … + to nterms =
sin
(ii) cos + cos( + ) + cos( + 2) + …. + to nterms =
2
n 1 n sin 2 2 2 n
cos
sin
(iii) cos. Cos2. Cos22 …. Cos(2n–1 ) =
sin 2 n 2 n sin
2
, n
= 1, = 2 k = – 1, = (2k + 1)
Trigonometric Equation 1.
General solution of the equations of the form
sin = 0
= n,
nI
(ii) cos = 0 = (2n +1) /2, n I
(iii) tan = 0
= n,
nI
(iv) sin = 1 = 2n +
(v) sin = 1
= 2n,
(iv) sin = –1
= n –
(i)
(vii) cos = –1
(ix) cos = tan (xi) sin2 = sin2 (xiii) tan2 = tan2 2.
2
or 2n +
= (2n +1), = n + = n = n
2
n I
,
3 2
(viii) sin = sin
= n +(–1)n = n + (x) tan = tan (xii) cos2 = cos2 = n
For general solution of the equation of the form:
acos + bsin = c, where c a 2 b 2 , divide both side by a 2 b 2 and put Thus the equation reduces to form cos ( – ) =
c a 2 b 2
a a 2 b 2
cos,
b a 2 b 2
= cos (say)
now solve using above formula. 3.
Some important points:
(i)
If while while solving an equation, we have to square it, then the roots roots found after squaring must must be checked wheather they satisfy the original equation or not.
(ii) If two equations equati ons are given then find the common common values values of between 0 & 2 and then add 2n to this common solution (value).
Inverse Trigonometric Function 1.
If y = sin x, then x = sin –1 y, similarly for other inverse T- functions
2.
Domain and Range of Inverse T- Function: Function
Domain (D)
Range (R)
16
3.
sin –1x
– 1 x 1
cos –1x
– 1 x 1
tan –1x
– < x <
cot –1x
– < x <
0
sec –1x
x – 1, x 1
0 ,
cosec –1x
x – 1, x 1
– , 0
–
2
2
0 –
2
2
2
2
2
Properties of Inverse T- Functions:
(i)
sin –1 (sin ) = provided – 2
2
tan –1 (tan ) = provided –
2
cos –1 (cos ) = provided
< <
cot –1 (cot ) = provided 0 < <
2
sec–1 (sec ) = provided 0 < or < 2
cosec –1 (cosec ) = provided –
2
< 0 or 0 <
2
2
(ii) sin (sin –1 x) = x provided – 1 x 1 cos (cos –1 x) = x provided – 1 x 1 tan (tan –1 x) = x provided – < x <
cot(cot –1 x) = x provided – < x <
sec (sec –1 x) = x provided – < x –1 or 1 x < cosec (cosec –1 x) = x provided – < x – 1 or 1 x < cos –1 (– x) = – cos –1 x
(iii) sin –1 (– x) = – sin –1 x,
tan –1 (– x) = – tan –1 x
cot –1 (– x) = – cot –1 x cosec –1 (– x) = – cosec –1 x sec –1 (– x) = – sec –1 x (iv) sin –1 x + cos –1 x = , x [– 1, 1] tan –1 x + cot –1 x = , x R 2
2
sec –1 x + cosec –1 x = 4.
2
, x (– , –1] [1, )
Value of one inverse function in terms of another inverse function :
(i)
sin –1 x = cos –1 1 x 2 tan 1
–1
–1
(ii) cos x = sin
(iii) tan –1 x = sin –1
2
1 x tan x 1 x2
x 1 x2
1 1 x
= cos –1
2
x
cot 1 cot 1
1 1 x2
1 x2 = sec –1 x x 1 x2
= cot –1
1 x
= sec –1
1
1 ,0 x x
1
1 1 = cosec –1 ,0 x x 1 x2
1
1 x2
= cosec –1
= sec –1 1 x 2 = cosec –1
1 x2 , x 0 x
1 (iv) sin –1 = cosec –1 x, x ( – , 1] [1, ) x 1 (v) cos –1 = sec –1 x, x ( – , 1] [1, ) x 1 cot 1 x for x 0 (vi) tan –1 – 1 x cot x for x 0 5.
Formulae for sum and difference of inverse trigonometric function :
(i)
x y ; if x > 0, y > 0, xy < 1 tan –1x + tan –1y = tan –1 1 xy
x y ; if x > 0, y > 0, xy > 1 (ii) tan –1x + tan –1y = + tan –1 1 xy
17
x y ; if xy > – 1 (iii) tan –1x – tan –1y = tan –1 1 xy x y ; if x > 0, y < 0, xy < –1 (iv) tan –1x – tan –1y = + tan –1 1 xy x y z xyz (v) tan –1x + tan –1y + tan –1z = tan –1 1 xy yz zx (vi) sin –1x sin –1y = sin –1 x 1 y 2 y 1 x 2 ; if x,y 0 & x2 + y2 1 (vii) sin –1x sin –1y = – sin –1 x 1 y 2 y 1 x 2 ; if x,y 0 & x2 + y2 > 1 (viii) cos –1x cos –1 y = cos –1 xy 1 x 2 1 y 2 ; if x,y > 0 & x 2 + y2 1 (ix) cos –1x cos –1 y = – cos –1 xy 1 x 2 1 y 2 ; if x,y > 0 & x2 + y2 > 1 6.
Inverse trigonometric ratios of multiple angles
2sin –1x = sin –1(2x 1 x 2 ), if – 1 x 1 2 x 2x (iii) 2tan –1x = tan –1 1 x 2 = sin –1 = cos –1 2 1 x
(ii) 2cos –1x = cos –1(2x2 –1), if – 1 x 1
(i)
1 x 2 1 x2
(iv) 3sin –1x = sin –1(3x – 4x 3)
(v) 3cos –1x = cos –1(4x3 – 3x)
3x – x 3 (vi) 3tan –1x = tan –1 2 1 3x
PROPERTIES & SOLUTION OF TRIANGLE Properties of triangle: 1.
A triangle has three sides and three angles. In any ABC, we write BC = a, AB = c, AC = b A
c
A
C
B a
B
b
C
and BAC = A, ABC = B, ACB = C 2.
In ABC : (i) A + B + C = (iii) a > 0, b > 0, c > 0
3.
(ii)
Sine formula: a b c = k (say) sin A sin B sin C
4.
b 2 c 2 a 2 2 bc
sin A sin B sin C = k (say) a b c
cos B =
c 2 a 2 b 2 2ac
cos C =
a 2 b 2 c 2 2ab
Projection formula:
A = b cos C + c cos B 6.
or
Cosine formula:
cos A = 5.
a + b > c, b + c > a, c + a > b
b = c cos A + a cos C
c = a cos B + b cos A
Napier’s Analogies:
18
tan 7.
tan
B C b c A cot 2 b c 2
CA ca B cot 2 ca 2
sin
A (s b)(s c) 2 bc
sin
C (s a)(s b) where 2s = a + b + c 2 bc
sin
B (s c)(s a ) 2 ca
(b) cos
A s(s a ) 2 bc
cos
B 2
(c)
A (s b)(s c) 2 s(s a )
tan
B (s c)(s a ) 2 s(s b)
tan
s(s b) ca
cos
C 2
s(s c) ab
tan
C (s b)(s a ) 2 s(s c)
,Area of triangle :
(i)
1 1 1 ab sin C = bc sin A = ca sin B 2 2 2
=
A B sc tan 2 2 s
tan
10.
Circumcircle of triangle and its radius :
(i)
tan
(ii) = s(s a ) (s b) (s c)
B C sa tan 2 2 s
9.
11.
tan
Half angled formula – In any ABC:
(a)
8.
A B a b C cot 2 a b 2
tan
a b c 2 sin A 2 sin B 2 sin C
R=
C A s b tan 2 2 s
(ii) R =
abc 4
Where R is circumradius
Incircle of a triangle and its radius :
(iii) r =
(iv) r = (s – a) tan
s
(v) r = 4R sin
A B C sin sin 2 2 2
A B C = (s – b) tan = (s – c) tan 2 2 2
(vi) cos A + cos B + cos C = 1 +
r R
B C A C B A sin b sin sin c sin sin 2 2 2 2 2 2 A B C cos cos cos 2 2 2
a sin
(vii) r =
12.
The radii of the escribed circles are given by :
(i)
, r 2 = , r 3 = sa s b sc
r 1 =
(iii) r 1 = 4R sin
(ii) r 1 = s tan
A B C cos cos , 2 2 2
r 2 = 4R cos
A B C sin cos , 2 2 2
(iv) r 1 + r 2 + r 3 – r = 4R (vi)
1 r 12
1 r 2 2
1 r 32
1 r 2
a 2 b 2 c 2
2
(viii) r 1r 2 + r 2r 3 + r 3r 1 = s2 (ix)
A B C , r 2 = s tan , r 3 = s tan 2 2 2
r 3 = 4R cos
(v)
1 1 1 1 r 1 r 2 r 3 r
(vii)
1 1 1 1 bc ca ab 2Rr
= 2R 2 sin A sin B sin C = 4Rr cos
A B C cos sin 2 2 2
A B C cos cos 2 2 2
B C C A A B cos b cos cos c cos cos 2 2 , r 2 2 , r 2 2 2 3 A B C cos cos cos 2 2 2
a cos
(x) r1 =
HEIGHT AND DISTANCE 1.
Angle of elevation and depression :
If an observer is at O and object is at P than XOP is called angle of elevation of P as seen from O.
19
Horizontal line
Q
P
Angle of depration
Angle of elevation
O
Horizontal line
X
If an observer is at P and object is at O, then QPO is called angle of depression of O as seen from P. 2.
Some useful result:
(i)
In any triangle ABC if AD : DB = m : n ACD = , BCD = & BDC = C b
c
B m D
A
B n
B
then (m + n) cot = m cot = ncot = ncotA – mcotB [m – n Theorem] (ii) d = h (cot – cot )
h
d
POINT 1.
Distance formula:
Distance between two points P(x 1, y1) and Q(x2, y2) is given by d(P, Q) = PQ = ( x 2 x1 ) 2 ( y 2 y1 ) 2 = Note:(i)
(Difference of x coordinate ) 2 ( Difference of y coordinate ) 2
d(p, Q) 0
(iii) d(P, Q) = d(Q, P) 2.
(ii) d(P,Q) = 0 P = Q (iv) Distance of a point (x, y) from origin (0, 0) =
x 2 y2
Use of Distance Formula : (a)
In Triangle: Calculate AB, BC, CA
(i)
If AB = BC = CA, then is equilateral.
(ii) If any two sides are equal then is isosceles.
(iii) If sum of square of any two sides is equal to the third, then is right triangle. (iv) Sum of any two equal to left third they do not form a triangle i.e. AB = BC + CA or BC = AC + AB or AC = AB + BC. Here points are collinear. (b)
In Parallelogram :
Calculate AB, BC, CD and AD. (i) If AB = CD, AD = BC, then ABCD is a parallelogram. (ii) If AB = CD, AD = BC and AC = BD, then ABCD is a rectangle. (iii) If AB = BC = CD = AD, then ABCD is a rhombus. (iv) If AB = BC = CD = AD and AC = BD, then ABCD (c)
For circumcentre of a triangle :
Circumcentre of a triangle is equidistant from vertices i.e. PA = PB = PC. Here P is circumcentre and PA is radius. (i) Circumcentre of an acute angled triangle is inside the triangle. (ii) Circumcentre of a right triangle is mid point of the hypotenuse.
20
(iii) Circumcentre of an obtuse angled triangle is outside the triangle. 3.
Section formula: (i)
Internally : AP m = , Here > 0 BP n m n P A (x 1 ,y1)
mx nx1 my 2 ny1 B (x1 ,y1 ) P , 2 m n m n
(ii) Externally : m n AP m = , BP n
P mx nx1 my 2 ny1 B (x1,y1) P , 2 m n m n
A (x1,y1)
x x y y (iii) Coordinates of mid point of PQ are 1 2 , 1 2 2 2 (iv) The line ax + by + c = 0 divides the line joining the points (ax by c) (x1,y1) & (x2, y2) in the ratio = – 1 (ax 2 by 2 c) (v) For parallelogram – midpoint diagonal AC = mid point of diagonal BD
x x x y y y (vi) Coordinates of centroid G 1 2 3 , 1 2 3 3 3 ax bx 2 cx 3 ay1 by 2 cy 3 (vii) Coordinates of incentre I 1 , a b c a b c (viii) Coordinates of orthocenter are obtained by solving the equation of any two altitudes. 4.
Area of Triangle:
The area of triangle ABC with vertices A(x 1, y1), B (x2, y2) and C(x3, y3). x1 1 = x 2 2 x3
=
1 2
y1 1 y 2 1 (Determinant method) y3 1
x1
y1
x2
y2
x3
y3
x1
y1
1 2
= [x1y2 + x2y3 + x3y1 – x2y1 – x3y2 – x1y3]
[Stair Method] Note:
(i)
Three points A, B, C are collinear if area of triangle is zero.
(ii) If in a triangle point arrange in anticlockwise then value of be +ve and if in clockwise then will be –ve. 5.
Area of Polygon:
Area of polygon having vertices (x 1, y1), (x2, y2), x3, y3) …….(xn, yn) is given by area x1 x2 1 x = 3 2 ... xn x1 6.
y1 y2 y3 . Points must be taken in order. .. . yn y1
Rotational Transformation:
If coordinates of any point P(x, y) with reference to new axis will be (x’, y’) then
21
x x' cos y' sin
7.
y sin cos
Some important points:
(i)
Three pts. A, B, C are collinear, if area of triangle is zero
(ii) Centroid G of ABC divides the median AD or BE or CF in the ratio 2 : 1 (iii) In an equilateral triangle, orthocenter, centroid, circumcentre, incentre coincide. (iv) Orthocentre, centroid and circumcentre are always collinear and centroid divides the line joining orthocenter and circumcentre in the ratio 2 : 1 (v) Area of triangle formed by coordinate axes & the line ax + by + c = 0 is
c2 . 2ab
STRAIGHT LINE 1.
Slope of a line:
The tangent of the angle that a line makes with +ve direction of the x-axis in the anticlockwise sense is called slope or gradient of the line and is generally denoted by m. Thus m = tan . (i) Slope of line || to x-axis is m = 0 (ii) Slope of line || to y-axis is m = (not defined) (iii) Slope of the line equally inclined with the axes is 1 or – 1 (iv) Slope of the line through the points A(x1, y1) and B(x2, y2) is (v) Slope of the line ax + by + c = 0, b 0 is –
y 2 y1 . x 2 x1
a b
(vi) Slope of two parallel lines are equal. (vii) If m1 & m2 are slopes of two lines then m 1m2 = – 1. 2.
Standard form of the equation of a line :
(i) (iii) (iv) (v) (vi)
Equation of x-axis is y = 0 (ii) Equation of y-axis is x = 0 Equation of a straight line || to x-axis at a distance b from it is y = b Equation of a straight line || to y-axis at a distance a from it is x = a Slope form : Equation of a line through the origin and having slope m is y = mx. Slope Intercept form : Equation of a line with slope m and making an intercept c on the y-axis is y = mx + c. (vii) Point slope form : Equation of a line with slope m and passing through the point (x 1, y1) is y – y1 = m(x – x1) y y1 x x1 (viii) Two points form : Equation of a line passing through the points (x 1, y1)&(x2, y2)is y 2 y1 x 2 x 1 (ix) Intercept form: Equation of a line making intercepts a and b respectively on x-axis and y-axis is (x) Parametric or distance or symmetrical form of the line: Equation of a line passing through x x1 y y1 making an angle , 0 , with the +ve direction of x-axis is =r 2 cos sin
x y 1. a b
(x 1, y1) and
x = x1 + r cos , y = y1 + r sin where r is the distance of any points P(x, y) on the line from the point (x 1, y1) (xi) Normal or perpendicular form : Equation of a line such that the length of the perpendicular from the origin on it is p and the angle which the perpendicular makes with the +ve direction of x-axis is , is x cos + y sin = p. 3.
Angle between two lines:
(i)
Two lines a1x + b1y + c1 = 0 & a 2x + b2y + c2 = 0 are (a)
Parallel if
a1 b1 c1 a 2 b 2 c 2
(b) Perpendicular if a 1a2 + b1 b2 = 0
22
(c)
Identical or coincident if
a1 b1 c1 a 2 b 2 c 2
(ii) Two lines y = m1x + c and y = m2x + c are (a) Parallel if m1 = m2 (c) 4.
If not above two, then = tan –1
(d) If not above three, then = tan –1
a 2 b l a 1 b 2 a 1a 2 b1 b 2
(b) Perpendicular if m 1m2 = –1
m1 m 2 1 m1m 2
Position of a point with respect to a straight line :
The line L(xi, yi) i = 1, 2 will be of same sign or of opposite sign according to the point A(x 1, y1) & B (x 2, y2) lie on same side or on opposite side of L (x, y) respectively. 5.
Equation of a line parallel (or perpendicular) to the line ax + by + c = 0 is ax + by + c’ = 0 (or bx – ay + = 0)
6.
Equation of st. lines through (x1,y1) making an angle
7.
length of perpendicular from (x1, y1) on ax + by + c = 0 is
8.
Distance between two parallel lines ax + by + ci = 0, i =1,2 is
9.
a 1 b1 Condition of concurrency for three straight lines L i ai x + bi y + ci = 0, i = 1, 2, 3 is a 2 b 2 a 3 b 3
10.
Equation of bisectors of angles between two lines: a 1x b1 y c1 a12 b12
11.
with y = mx + c is y –y1 =
m tan (x – x1) 1 m tan
| ax1 by1 c a 2 b 2 | c1 c 2 | a 2 b 2 c1 c2 = 0 c3
a 2 x b 2 y c 2 a 22 b 22
Family of straight lines :
The general equation of family of straight line will be written in one parameter The equation of straight line which passes through point of intersection of two given lines L 1 and L2 can be taken as L1 + L2= 0 12. Homogenous equation: If y = m 1x and y = m 2x be the two equations represented by ax2 + 2hxy + by2 =, 0 then m1+ m2 = – 2h/b and m1m2 = a/b 13.
General equation of second degree: a h g ax + 2hxy + by + 2gx + 2fy + c = 0 represents a pair of straight line if = h b f = 0 g f c 2
2
If y = m1x + c & y = m2x + c represents two straight lines then m 1+ m2 = 14.
2h b
, m1m2 =
a . b
Angle between pair of straight lines:
The angle between the lines represented by ax 2 + 2hxy + by2 + 2gx + 2fy + c = 0 or ax 2 + 2hxy + by2 = 0 is tan = 2 h 2 ab (a b)
The two lines given by ax 2 + 2hxy + by2 = 0 are (a) Parallel and coincident iff h2 – ab = 0 (b) Perpendicular iff a + b = 0 2 2 (ii) The two line given by ax + 2hxy + by + 2gx + 2fy + c = 0 are (a) Parallel if h2 – ab = 0 & af 2 = bg2(b) Perpendicular iff a + b = 0 2 (c) Coincident iff g – ac = 0 13. Combined equation of angle bisector of the angle between the lines ax2 + 2hxy + by2 = 0 (i)
x 2 y 2 xy a b h
Circle 1.
General equation of a circle: x2 + y2 + 2gx + 2fy + c = 0 where g, f and c are constants
23
(i)
1 1 Centre of the circle is (–g, – f) i. e coeff . of x, coeff . of y 2 2
(ii) Radius is g 2 f 2 c 2.
3. 4.
Central (Centre radius) form of a circle:
(i) (x – h) 2 + (y – k) 2 = r 2 , where (h, k)is circle centre and r is the radius (ii) x2 + y2 = r 2 , where (0 ,0) origin is circle centre and r is the radius Diameter form : If (x1,y1) and (x2, y2) are end pts. Of a diameter of a circle, then its equation is (x – x1) (x – x 2) + (y – y1) (y – y2) = 0 Parametric equation :
(i)
The parametric equation of the circle x2 + y2 = r 2 are x = r cos , y = rsin , where point (r cos, r sin )
(ii) The parametric equations of the circle (x – h) 2 + (y – k)2 = r 2 are x = h + rcos, y = k + rsin (iii) The parametric equation of the circle x2 +y2 + 2gx + 2fy + c = 0 are x = – g + g 2 f 2 c cos , y = – f + g 2 f 2 c sin (iv) For circle x2 + y2 = a2 = a2, equation of chord joining 1& 2 is x cos = r cos 5.
1 2 2
+ y sin
1 2 2
1 2 2
Concentric circles: Two circles having same centre C (h,k) but different radii r 1 & r 2 respectively are called concentric
circles. 6.
Position of a point w.r.t. a circle: A point (x1,y1) lies outside, on or inside a circle
S x2 + y2 + 2gx + 2fy + c = 0 according as S1 x12 y12 2gx1 2fy1 c is +ve, zero or – ve 7.
Chord length (length of intercept) = 2 r 2 p 2
8.
Intercepts made on coordinate axes by the circle:
(i)
x axis = 2 g 2 c
(ii) y axis = 2 f 2 c
9.
Length of tangent = S1
10.
Length of the intercept made by line: y = mx + c with the circle x 2 + y2 = a2 is 2
a 2 (1 m 2 ) c 2 1 m
2
or (1 + m2 ) | x1 – x2|
where | x1 – x2| = difference of roots i.e.
D . a
11.
Condition of Tangency: Circle x2 + y2 = a2 will touch the line y = mx + c if c = a 1 m 2 .
12.
Equation of tangent, T = 0:
Equation of tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 at any point (x 1, y1) is xx1 + yy1 + g(x + x1) + f( y + y1) + c = 0 (ii) Equation of tangent to the circle x2 + y2 = a2 at any point (x1, y1) is xx 1 + yy1 = a2 (iii) In slope form : From the condition of tangency for every value of m. (i)
The line y = mx a 1 m 2 is a tangent to the circle x 2 + y2 = a2 and its point of contact is
am a , 1 m 2 1 m 2 (iv) Equation of tangent at (a cos , a sin ) to the circle x2 + y2 = a2 is x cos + y sin = a. 13.
Equation of normal :
(i)
Equation of normal to the circle x2 + y2 + 2gx + 2fy + c = 0 at any point P(x 1, y1) is y f y – y1= 1 (x – x1) x1 g
24
(ii) Equation of normal to the circle x2 + y2 = a2 at any point (x 1,y1) is xy1 – x1y = 0 14.
Equation of pair of tangents SS1 = T2:
15.
The point of intersection of tangents drawn to the circle x2 + y2 = r 2 at point 1 & 2 is given as
r cos 1 2 r cos 1 2 2 , 2 cos 1 2 cos 1 2 2 2 16.
Equation of the chord of contact of the tangents drawn from point P outside the circle is T = 0
17.
Equation of a chord whose middle pt. is given by T = S1
18.
Director circle : Equation of director circle for x 2 + y2 = a2 is x2 + y2 = 2a2. Director circle is a concentric circle whose
radius is
2 times the radius of the given circle.
19.
Equation of polar of point(x1,y1) w.r.t. the circle S = 0 is T = 0
20.
a 2 l a 2 m Coordinates of pole : Coordinates of pole of the line lx + my + n = 0 w.r.t the circle x 2 + y2 = a2 are n , n
21.
Family of circles:
(i)
S + S’ = 0 represents a family of circles passing through the pts. of intersection of
S=0
& S’ = 0 if – 1 (ii) S + L = 0 represents a family of circles passing through the point of intersection of
S=0
&L=0 (iii) Equation of circle which touches the given straight line L = 0 at the given point (x1, y1) is given as (x – x1)2 + (y – y1)2 + L = 0. (iv) Equation of circle passing through two points A(x 1,y1) & B(x2, y2) is given as (x – x1) (x – x2) + (y – y1) (y – y2) +
x x1 x2
y 1 y1 1 0. y2 1
22.
Equation of Common Chord is S – S1 = 0.
23.
The angle of intersection of two cir cles with centres C1 & C2 and radii r 1 & r 2 is given by
cos =
r 12 r 12 d 2 , where d = C1C2 2 r 1r 2
24.
Position of two circles : Let two circle with centres C1, C2 and radii r 1, r 2.
Then following cases arise as (i)
C1C2 > r 2 + r 2 do not intersect or one outside the other, 4 common tangents.
(ii) C1C2 = r 1 + r 2 Circles touch externally, 3 common tangents. (iii) | r 1 – r 2| < C1 C2 < r 1 + r 2 Intersection at two real points, 2 common tangents. (iv) C1 C2 = |r 1 – r 2| internal touch, 1 common tangent. (v) C1 C2 < |r 1 + r 2| one inside the other, no tangent. Note:Point of contact divides C1 C2 in the ratio r 1 : r 2 internally or externally as the case may be 25.
Equation of tangent at point of contact of circle is S1 – S2 = 0
26.
Radical axis and radical centre :
(i) Equation of radical axis is S – S1 = 0 (ii) The point of concurrency of the three radical axis of three circles taken in pairs is called radical centre of three circles. 27.
Orthogonality condition:
If two circles S x2 + y2 + 2gx + 2fy + c = 0 and S’ = x 2 + y2 + 2g’x + 2f’y + c’ = 0 intersect each other orthogonally, then 2gg’ + 2ff’ = c + c’.
PARABOLA
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1.
Standard Parabola : 2
2
2
2
Imp. Terms
y = 4ax
y = – 4ax
x = 4ay
x = – 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
x+a
a–x
y+ a
a–y
(at2, 2at)
(– at2, 2at)
(2at, at2)
(2at, – at2)
Parametric
x = at2
x = – at2
x = 2at
x = 2at
Equations
y = 2at
y = 2at
y = 2at 2
y = – at2
Distance Parametric Coordinates
Y
a – = x
Directrix
Tangent at the vertex (a,2a) Latus Rectum Focus x S(a,0)
A
Vertex (0,0)L
axis of the (a,–2a) parabola i.e. y =0
y2 = 4ax
(–a,2a) L (–a,0)
Directrix A
S x=a axis of the L(–a,–2a) parabola Latus Rectum y2 = – ax
Y Latus Rectum (–a,2a) L’
Focus S (0,a)
L (a,2a)
A
x Tangent at (0,b)the vertex y=–a i.e. y =0
26
x2 = 4ay
y=a A
x (a– 2a)
(–a,–2a) L’
S
L (a– 2a)
Latus Rectum Y 2
x = – 4ay 2.
Special form of Parabola
Parabola which has vertex at (h, k), latus rectum
and axis parallel to x-axis is (y – k) 2 =
and axis parallel to y-axis is (x – h) 2 =
(x – h)
axis is y = k and focus at h , k
4
Parabola which has vertex at (h, k), latus rectum
(y – k)
axis is x = h and focus at h, k
4
Equation of the form ax 2 + bx + c = y represents parabola. 2
i.e. y –
b 4ac b 2 4ac b 2 b and axes parallel to y-axis a x ,with vertex , 4a 2 a 2 a 4 a
Note : Parametric equation of parabola (y – k) 2 = 4a(x – h) are x = h + at 2, y = k + 2at 3.
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 y12 – 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.
Equations of tangent in different forms: (i)
Point form / Parametric form Equations of tangent of all other standard parabolas at
Equation of parabola
Tangent at (x1, y1)
Parametric coordinates ‘t’
(x 1, y1) / at t (parameter)
Tangent of ‘t’
y2 = 4ax
yy1 = 2a(x + x1)
(at2, 2at)
ty = x + at2
y2 = – 4ax
yy1 = – 2a(x + x1)
(– at2, 2at)
ty = – 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
Point of contact in
Equations of tangent in
Condition of
parabolas
terms of slops(m)
terms of slope (m)
Tangency
y2 = 4ax
a 2a 2 ' m m
y = mx +
a m
c=
y2 = – 4ax
a 2a 2 ' m m
y = mx –
a m
c = –
x2 = 4ay
(2am, am2)
y = mx – am2
a m a m
c = – am2
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x2 = – 4ay 5. 6. 7.
(–2am, –am2)
y = mx + am2
c + am2
Point of intersection of tangents at any two points P(at 12, 2at1) and Q(at22, 2at2) on the parabola y2 = 4ax is (at1t2, a(t1 +
t2)) i.e. (a(G.M.)2, a(2A.M.)) Combined equation of the pair of tangents drawn from a point to a parabola is SS’ = T 2, where S = y2 – 4ax, S’ = y12 – 4ax1 and T = yy1 – 2a(x + x1) Equations of normal in different forms (i) Point form / Parametric form Equations of normals of all other standard parabolas at (x 1, y1) / at t (parameter) Equation of
Normal at
parabola
(x1, y1)
2
y = 4ax
y–y=
y2 = – 4ax x2 = 4ay
y– y =
y1 2a
x2 = – 4ay
at ‘t’
2
(x – x1)
y1 (x – x1) 2a
y – y = –
Normals
Point ‘t’
2a (x – x1) x1
2a y– y = (x – x1) x1
(at , 2at)
y + tx = 2at + at3
(–at2, 2at)
y-tx = 2at + at3
(2at, at2)
x + ty = 2at + at3
(2at, – at2)
x – ty = 2at + at3
(ii) Slope form Equations of normal, point of contact, and condition of normality in terms of slope (m) Equation of
Point of
parabola
contact
2
Condition of Normality
y = 4ax
(am , –2am)
y = mx – 2am –am
c = –2am –am3
y2 = – 4ax
(–am2, 2am)
y = mx + 2am +am3
c = am + am3
x2 = 4ay
2a a , 2 m m
y = mx + 2a +
2a a , 2 m m
y = mx – 2a –
x2 = – 4ay
2
Equations of normal 3
a 2
c = 2a +
2
c = –2a –
m a m
a m2 a m2
Note:
8.
(i) In circle normal is radius itself. (ii) Sum of ordinates (y coordinate) of foot of normals through a point is zero. (iii) The centroid of the triangle formed by taking the foot of normals as a vertices of concurrent normals of y 2 = 4ax lies on x-axis. Condition for three normals from 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 t 2 then t2 = – t1 –
2 t1
(ii) The normals to y2 = 4ax at t1 and t2 intersect each other at the same parabola at t3, then and t3 = – t1 – t2 10.
(i)
Equation of focal chord of parabola y2 = 4ax at t1 is y =
2 t1 t12 1
t1t2 = 2
(x – a)
If focal chord of y2 = 4ax cut (intersect) at t1 and t2 then t1t2 = – 1 (t 1 must not be zero) (ii) Angle formed by focal chord at vertex of parabola is tan =
2 |t2 – t1| 3
(iii) Intersecting point of normals at t1 and t2 on the parabola y2 = 4ax is (2a + a(t12 + t22 +t1t2), – at1t2 (t1 + t2)) 11. Equation of chord of parabola y2 = 4ax which is bisected at (x 1, y1) is given by T = S1
28
12.
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 (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
n 2am x + my + n = 0 w.r.t. the parabola y2 = 4ax is ,
14.
Diameter : It is locus of mid point of set of parallel chords and equation is given by T = S 1
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 from focus to any tangent will be tangent at vertex. (iii) If tangents drawn at ends point of a focal chord are mutually perpendicular then their point of intersection will lie on directrix. (iv) Any light ray traveling parallel to axis of the parabola will pass through focus after reflection through parabola. (v) Angle included between focal radius of a point and perpendicular from a point to directrix 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 tangency and directrix will make right angle at focus. (vii) Circle drawn on any focal radius as diameter will touch tangent at vertex. (viii) Circle drawn on any focal chord as diameter will touch directrix.
ELLIPSE 1.
Standard Ellipse (e < 1)
x 2 y 2 2 2 1 a b
Ellipse Imp.terms For a > b
For b > a
Centre
(0,0)
(0,0)
Vertices
( a,0)
(0, b)
Length of major axis
2a
2b
Length of minor axis
2b
2a
Foci
(ae, 0)
(0, be)
Equation of directrices
x = a/e 2
2
y b/e 2
Relation in a, b and c
b = a (1– a )
a2 = b2 (1– e2 )
Length of latus rectum
2b2/a
2a2/b
Ends of latus rectum
2 ae, b a
a 2 , be b
Parametric coordinates
(a cos , b sin)
( acos , b sin) 0 < 2
Focal radii Sum of focal radii
SP = a – ax1
SP = b – ey1
S’P = a + ex 1
S’P = b + ey1
SP + S’P =
2a 2b
n
Distance bt foci
2ae
2be
Distance btn directrices
2a/e
2b/e
Tangent at the vertices
x = – a, x = a
y = b, y = –b
Note: If P is any point on ellipse and length of perpendiculars from to minor axis and major axis are p 1 & p2, then |x p| 2
= p1, |y p| = p2
2
p12 p 22 =1 a
b
29
Y x i r t c e r i DZ’
X’
p(x,y)
(0,b)
A’ (–a,0)
M
C S’(–ae,0) S (ae,0)
x i r t c e r i D
Z
x
A (a,0)
(0,–b) B’ Y’
x= –a/e
x=a/e
a>b
y =b/c
k
Z A(0,b)
S X’
B’ (–a,0)
) e b , 0 (
X
C (0,0) B (a,0) S’ A’(0,– b) Z’
2.
Special form of ellipse:
If the centre of an ellipse is at point (h, k) and the directions of the axes are parallel to the coordinate axes, then its equation is 3.
(x h ) 2 a2
( y k ) 2 b 2
1.
Auxillary Circle: The circle described by taking centre of an ellipse as centre and major axis as a diameter is called an
auxillary circle of the ellipse. If
x2 a
2
y2 b
2
= 1 is an ellipse then its auxillary circle is x2 + y2 = a2.
Note: Ellipse is locus of a point which moves in such a way that it divides the normal of a point on diameter of a point of
circle in fixed ratio. 4.
5.
Position of a point and a line w.r.t. an ellipse: x12
y12
The point lies outside, on or inside the ellipse if S1 =
The line y = mx + c does not intersect, touches, intersect, the ellipse if a2m2 + b2 < = > c2
a2
b 2
–1 >, = or < 0
Equation of tangent in different forms : (i)
Point form : The equation of the tangent to the ellipse
(ii) Slope form : If the line y = mx + c touches the ellipse
x2 a
2
x2 a
2
y2 b
2
y2 b
2
= 1 at the point (x1, y1) is
xx1 a
2
yy1 b 2
1.
= 1, then c2 = a2m2 + b2. Hence, the straight line y
= mx a 2 m 2 b 2 always represents the tangents to the ellipse. Point of contact :
Line y = mx a 2 m 2 b 2 touches the ellipse
x2 a2
a 2 m b 2 . = 1 at , 2 2 2 2 2 2 2 b a m b a m b y2
(iii) Parametric form : The equation of tangent at any point (a cos , b sin ) is
x y cos + sin = 1. a b
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