Chapter wise summary & formulas

➢ 𝑒 = 2.7185
Chapter #1 Note that
Hyperbolic Function
i. (
cosh − 1 hx = ln x + x 2 + 1 )
e x − e−x ii. sinh − 1 hx = ln (x + x2 − 1)
❖ sin h( x) =
2
❖ cosh( x) =
e + e−x
x Chapter #2
𝑑
2 i. 𝐶 = 0 , C is constant
𝑑𝑥
sin hx e x − e − x
❖ tan h ( x ) = = ii.
𝑑
𝐶 𝑓 (𝑥 ) = 𝐶
𝑑
𝑓 (𝑥 )
coshx e x + e − x 𝑑𝑥 𝑑𝑥
2 2 𝑑 𝑑 𝑑
❖ cosh x − sinh x = 1 iii. (𝑓 (𝑥 ) ± g(𝑥 )) = 𝑓 (𝑥 ) ± g (𝑥 )
𝑑𝑥 𝑑𝑥 𝑑𝑥
2 2 𝑑 𝑑 𝑑
❖ 1 − tanh x = sec h x iv. (𝑓 ∙ g ) = 𝑓 g+g 𝑓 (product rule)
𝑑𝑥 𝑑𝑥 𝑑𝑥
❖ coth 2 x − 1 = cos ech2 x g
d
f −f
d
g
d f
❖ If 𝑓 (−𝑥 ) = −𝑓 (𝑥 ) ⇒f(x) is odd function v.   = dx dx (Quotient rule)
❖ If 𝑓 (−𝑥 ) = +𝑓 (𝑥 ) ⇒f(x) is even function dx  g  g2
❖ If 𝑓 (−𝑥 ) ≠ ±𝑓 (𝑥 ) ⇒f(x) is neither even nor 𝑑 𝑑
vi. [𝑓 (𝑥 )]𝑛 = 𝑛[𝑓(𝑥 )]𝑛−1 𝑓 (𝑥 )
odd 𝑑𝑥 𝑑𝑥
❖ 𝑥 = 𝑎𝑡 2 , 𝑦 = 2𝑎𝑡 and 𝑦 = 𝑎𝑡 2 , 𝑥 = 2𝑎𝑡 (Power Rule)
represent parabola 𝑦 2 = 4𝑎𝑥 and 𝑥 2 = 4𝑎𝑦 vii. If 𝑦 = 𝑓(𝑥 ), 𝑧 = g(𝑥 ) then
respectively. 𝑑𝑦 𝑑𝑦 𝑑𝑥
= ∙ (chain rule)
❖ 𝑥 = 𝑎𝑐𝑜𝑠𝜃 , 𝑦 = 𝑎𝑠𝑖𝑛𝜃 𝑑𝑧 𝑑𝑥 𝑑𝑧
𝑥2 𝑦2 Derivation of trigonometric function
2 + =1
represent ellipse 𝑑 𝑑
𝑎 𝑏2 i. (𝑠𝑖𝑛𝑥) = 𝑐𝑜𝑠𝑥 𝑥
❖ 𝑥 = 𝑎𝐶𝑜𝑠𝜃 , 𝑦 = 𝑎𝑆𝑖𝑛𝜃 𝑑𝑥 𝑑𝑥
𝑑 𝑑
represent circle 𝑥 2 + 𝑦 2 = 𝑎 2 ii. (𝑐𝑜𝑠𝑥) = −𝑠𝑖𝑛𝑥 𝑥
𝑑𝑥 𝑑𝑥
❖ 𝑥 = 𝑎𝑆𝑒𝑐𝜃 , 𝑦 = 𝑎𝑡𝑎𝑛𝜃 𝑑 𝑑
𝑥2 𝑦2 iii. (𝑡𝑎𝑛𝑥) = 𝑠𝑒𝑐2𝑥 𝑥
𝑑𝑥 𝑑𝑥
represent Hyperbola
𝑎 2 − 𝑏2
=1 𝑑 𝑑
Some important results: iv. (𝑐𝑜𝑡𝑥) = −𝑐𝑜𝑠𝑒𝑐2𝑥 𝑥
𝑑𝑥 𝑑𝑥
𝑑 𝑑
v. (𝑠𝑒𝑐𝑥) = 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥 𝑥
xn − an 𝑑𝑥 𝑑𝑥
i. lim = na n−1 𝑑 𝑑
x→c x − a vi. (𝑐𝑜𝑠𝑒𝑐𝑥 ) = −𝑐𝑜𝑠𝑒𝑐𝑥𝑐𝑜𝑡𝑥 𝑥
𝑑𝑥 𝑑𝑥
ax −1 Derivation of inverse Trigonometric
ii. lim = loge a = ln a
x →0 x function
ex − 1 𝑑
(𝑠𝑖𝑛−1 𝑥 ) =
1 𝑑
𝑥
iii. lim = ln e = 1 i.
x →0 x 𝑑𝑥 √1−𝑥 2 𝑑𝑥
n 𝑑 −1 𝑑
 1 ii. (𝑐𝑜𝑠 −1 𝑥) = 𝑥
iv. lim 1 +  = e 𝑑𝑥 √1−𝑥 2 𝑑𝑥
n →
 n 𝑑 1 𝑑
iii. (𝑠𝑒𝑐 −1 𝑥 ) = 𝑥
v. lim(1 + n ) = e
1
n 𝑑𝑥 𝑥√𝑥 2 −1 𝑑𝑥
n →0 𝑑 −1 𝑑
sin  iv. (𝑐𝑜𝑠𝑒𝑐 −1 𝑥 ) = 𝑥
vi. lim =1 𝑑𝑥 𝑥√𝑥 2 −1 𝑑𝑥
 →0  𝑑 1 𝑑
v. (𝑡𝑎𝑛−1 𝑥 ) = 𝑥
➢ 𝑒 ∞ = ∞ and 𝑒 −∞ = 0 𝑑𝑥 1+𝑥 2 𝑑𝑥
𝑑 1 𝑑
➢ 𝑒0 = 1 vi. (𝑐𝑜𝑡 −1 𝑥) = 𝑥
𝑑𝑥 1+𝑥 2 𝑑𝑥
 Derivation of Hyperbolic Functions f ( x)
i.
𝑑
𝑠𝑖𝑛ℎ𝑥 = 𝑐𝑜𝑠ℎ𝑥 𝑥
𝑑  f ( x)
dx = ln f ( x) + c
𝑑𝑥 𝑑𝑥
𝑑 𝑑 x n+1
ii. 𝑐𝑜𝑠ℎ𝑥 = 𝑠𝑖𝑛ℎ𝑥 𝑥  x dx = +c
n
i.
𝑑𝑥 𝑑𝑥 n +1
𝑑 𝑑
iii. 𝑡𝑎𝑛ℎ𝑥 = 𝑠𝑒𝑐ℎ2 𝑥 𝑥 ii. ∫ 𝑐𝑜𝑠𝑥𝑑𝑥 = 𝑠𝑖𝑛𝑥 + 𝑐
𝑑𝑥 𝑑𝑥
𝑑 2 𝑑 iii. ∫ 𝑠𝑒𝑐𝑥𝑑𝑥 = 𝑙𝑛|𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥 | + 𝑐
iv. 𝑐𝑜𝑡ℎ𝑥 = −𝑐𝑜𝑠𝑒𝑐ℎ 𝑥 𝑥
𝑑𝑥 𝑑𝑥 iv. ∫ 𝑐𝑜𝑠𝑒𝑐𝑥𝑑𝑥 = 𝑙𝑛|𝑐𝑜𝑠𝑒𝑐𝑥 − 𝑐𝑜𝑡𝑥 | + 𝑐
𝑑 𝑑
v. 𝑠𝑒𝑐ℎ𝑥 = − 𝑠𝑒𝑐ℎ𝑥 𝑡𝑎𝑛ℎ𝑥 𝑥 v. ∫ 𝑡𝑎𝑛𝑥𝑑𝑥 = 𝑙𝑛|𝑠𝑒𝑐𝑥 | = −𝑙𝑛|𝑐𝑜𝑠𝑥 | + 𝑐
𝑑𝑥 𝑑𝑥
𝑑 𝑑 vi. ∫ 𝑐𝑜𝑡𝑥𝑑𝑥 = 𝑙𝑛|𝑠𝑖𝑛𝑥 | = −𝑙𝑛|𝑐𝑜𝑠𝑒𝑐𝑥 | + 𝑐
vi. 𝑐𝑜𝑠𝑒𝑐ℎ𝑥 = − 𝑐𝑜𝑠𝑒𝑐ℎ𝑥 𝑐𝑜𝑡ℎ𝑥 𝑥
𝑑𝑥 𝑑𝑥 vii. ∫ 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥𝑑𝑥 = 𝑠𝑒𝑐𝑥 + 𝑐
Inverse hyperbolic function viii. ∫ 𝑐𝑜𝑠𝑒𝑐𝑥𝑐𝑜𝑡𝑥𝑑𝑥 = −𝑐𝑜𝑠𝑒𝑐𝑥 + 𝑐
𝑑 1 𝑑
i. 𝑠𝑖𝑛ℎ−1 𝑥 = ix. ∫ 𝑠𝑒𝑐 2 𝑥𝑑𝑥 = 𝑡𝑎𝑛𝑥 + 𝑐
𝑑𝑥 √1+𝑥 2 𝑑𝑥
𝑑 1 𝑑
x. ∫ 𝑐𝑜𝑠𝑒𝑐 2𝑥𝑑𝑥 = −𝑐𝑜𝑡𝑥 + 𝑐
ii. 𝑐𝑜𝑠ℎ−1 𝑥 = 1−𝑐𝑜𝑠2𝑥
𝑑𝑥 √𝑥 2 −1 𝑑𝑥 xi. ∫ 𝑠𝑖𝑛2 𝑥𝑑𝑥 = ∫ ( 2 ) 𝑑𝑥
𝑑 1 𝑑
iii. 𝑡𝑎𝑛ℎ−1 𝑥 = 1+𝑐𝑜𝑠2𝑥
𝑑𝑥 1−𝑥 2 𝑑𝑥 xii. ∫ 𝑐𝑜𝑠 2 𝑥𝑑𝑥 = ∫ ( 2 ) 𝑑𝑥
𝑑 1 −1 𝑑
iv. 𝑐𝑜𝑡ℎ−1 𝑥 = or xiii. ∫ 𝑡𝑎𝑛2 𝑥𝑑𝑥 = ∫(𝑠𝑒𝑐 2 𝑥 − 1)𝑑𝑥
𝑑𝑥 1−𝑥 2 𝑥 2 −1 𝑑𝑥
𝑑 −1 𝑑
v. 𝑠𝑒𝑐ℎ−1 𝑥 = xiv. ∫ 𝑐𝑜𝑡 2 𝑥𝑑𝑥 = ∫(𝑐𝑜𝑠𝑒𝑐 2 𝑥 − 1)𝑑𝑥
𝑑𝑥 𝑥√1−𝑥 2 𝑑𝑥
𝑑 −1 −1 𝑑 xv. ∫ 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝑐
vi. 𝑐𝑜𝑠𝑒𝑐ℎ 𝑥 =
𝑑𝑥 |𝑥|√1+𝑥 2 𝑑𝑥 𝑎𝑥
𝑑 −1 𝑑 xvi. ∫ 𝑎 𝑥 𝑑𝑥 = +𝑐
vii. 𝑐𝑜𝑠𝑒𝑐ℎ−1 𝑥 = 𝑙𝑛𝑎
𝑑𝑥 𝑥√1+𝑥 2 𝑑𝑥
Note
Derivation of Exponential Function
𝑑 𝑑𝑥 ❖ In case of ∫ 𝑠𝑖𝑛4 𝑥𝑑𝑥 or ∫ 𝑐𝑜𝑠 4 𝑥𝑑𝑥
i. ( 𝑒 𝑥 ) = 𝑒𝑥
𝑑𝑥 𝑑𝑥 we use double angle identity two times
𝑑 𝑑𝑥
ii. (𝑎 ) = 𝑎
𝑥 𝑥
𝑙𝑛 𝑎 Some useful substitution
𝑑𝑥 𝑑𝑥
𝑑 1 𝑑𝑥
iii. (𝑙𝑛𝑥) = i. For √𝑎2 − 𝑥 2 , we put
𝑑𝑥 𝑥 𝑑𝑥
𝑥 = 𝑎𝑠𝑖𝑛𝜃 𝑜𝑟 𝑥 = 𝑎𝑐𝑜𝑠𝜃 or 𝑥 = 𝑎𝑡𝑎𝑛ℎ𝜃
𝑑 1 𝑑𝑥
iv. log𝑎 𝑥 = ii. For √𝑥 2 − 𝑎2 , 𝑤𝑒 𝑝𝑢𝑡 𝑥 = 𝑎𝑠𝑒𝑐𝜃 or 𝑥 =
𝑑𝑥 𝑥𝑙𝑛𝑎 𝑑𝑥
Maclaurin Series 𝑎𝑐𝑜𝑠𝑒𝑐𝜃 Or 𝑥 = 𝑎𝑐𝑜𝑠ℎ𝜃
x2 x3 iii. For √𝑥 2 + 𝑎2 , we put
f ( x) = f (0) + xf (0) + f (0) + f (0) + ...
2! 3! 𝑥 = 𝑎𝑡𝑎𝑛𝜃 𝑜𝑟 𝑥 = 𝑎𝑐𝑜𝑡𝜃 Or 𝑥 = 𝑎𝑠𝑖𝑛ℎ𝜃
 n
f (0) n
= x An important rule
n =0 n! ❖ ∫ 𝑒 𝑎𝑥 [𝑎𝑓 (𝑥 ) + 𝑓 ′(𝑥 )]𝑑𝑥 = 𝑒 𝑎𝑥 𝑓 (𝑥 ) + 𝑐
Tayler Series ❖ ∫ 𝑒 𝑥 [𝑓 (𝑥 ) + 𝑓 ′(𝑥 )]𝑑𝑥 = 𝑒 𝑥 𝑓(𝑥 ) + 𝑐
( x − a) 2 ( x − a )3 Properties of definite integral
f ( x) = f (a) + ( x − a) f (a) + f (a) + f  (a) +
2! 3! a

..... +
( x − a) n n
f (a) + ...
❖  f ( x)dx = 0
n! a
b a

Chapter #3 ❖ 
a
f ( x)dx = −  f ( x)dx
b
Rule I
 f (x )n+1 + c , 𝑛 ≠ −1
c b c

  f ( x ) f n
( x ) dx =
n +1
❖ 
a
f ( x)dx =  f ( x)dx +  f ( x)dx
a b

Rule II where 𝑎 < 𝑏 < 𝑐

 ❖ 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) (point-slope form)
Chapter #4 𝑦 −𝑦
❖ 𝑦 − 𝑦1 = 𝑥2 −𝑥1 (𝑥 − 𝑥1 ) (2 point form)
2 1
𝑥 𝑥
Distance formula ❖ + = 1 (2 intercepts form)
𝑎 𝑏
|̅̅̅̅
𝐴𝐵| = 𝑑 = √(𝑥2 −𝑥1 )2 +(𝑦2 −𝑦1)2 ❖ 𝑥𝑐𝑜𝑠𝛼 + 𝑦𝑠𝑖𝑛𝛼 = 𝑃 (normal form)
𝑥−𝑥1 𝑦−𝑦1
❖ If point 𝑃(𝑥, 𝑦) divide segment AB internally ❖ = = r (parametric form or
𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃
with 𝑘1 : 𝑘2 then coordinates of P are symmetric form)
𝑘1 𝑥2 +𝑘2 𝑥1 ❖ Distance of a point 𝑃(𝑥1, 𝑦1 ) from line
𝑥=
𝑘1 +𝑘2 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 is given by
𝑘1 𝑦2 +𝑘2 𝑦1 |𝑎𝑥1 +𝑏𝑦1 +𝑐|
𝑦= 𝑑=
𝑘1 +𝑘2 √𝑎2 +𝑏2
❖ If point 𝑃 (𝑥, 𝑦) divide AB externally with ❖ If three vertices 𝐴(𝑥1 , 𝑦1 ) , 𝐵(𝑥2 , 𝑦2 ) and
𝑘1 : 𝑘2
𝐶 (𝑥3 , 𝑦3 ) then
𝑘1 𝑥2 −𝑘2 𝑥1 𝐾1 𝑦2 −𝑘2𝑦1
𝑥= ,𝑦= 𝑥1 𝑦1 1
𝑘1 −𝑘2 𝑘1 −𝑘2 1
Area of ∆ = |𝑥2 𝑦2 1|
2
Mid point formula 𝑥3 𝑦3 1
𝑥1 +𝑥2 𝑦1 +𝑦2 ➢ When points are concurrent then area is zero
𝑥= ,𝑦= 𝑥1 𝑦1 1
2 2
❖ The point of intersection median is called & then 2 𝑦2 1| = 0
| 𝑥
centroid. The centroid of ∆ 𝑥3 𝑦3 1
𝑚 −𝑚
𝑥1 +𝑥2 +𝑥3 ❖ Angle from first to 2nd line 𝑡𝑎𝑛𝜃 = 2 1
𝑥= 1+𝑚1 𝑚2
3 ❖ The angle 𝜃 between the pair of straight lines
𝑦1 +𝑦2 +𝑦3
𝑦= represented by 𝑎𝑥 2 + 2ℎ𝑥𝑦 + 𝑏𝑦 2 = 0 is
3
❖ The point of intersection of angle bisector is 2√ℎ2−𝑎𝑏
given by 𝑡𝑎𝑛𝜃 =
called incentre. The incentre of ∆ 𝑎+𝑏

𝑥=
𝑎𝑥1 +𝑏𝑥2 +𝑐𝑥3
𝑎+𝑏+𝑐
,𝑦=
𝑎𝑦1 +𝑏𝑦2 +𝑐𝑦3
𝑎+𝑏+𝑐 Chapter #6
❖ If axis are shifted to new new origin 𝑂′ (ℎ, 𝑘) It should be noted that
then the coordinates of 𝑃 (𝑥, 𝑦) and 𝑃(𝑋, 𝑌) ❖ If 𝑒 = 1 then conic is Parabola.
are related as ❖ If 0 < e < 1 then conic is ellipse.
𝑥 = 𝑋 + ℎ ⇒ 𝑋 = 𝑥 − ℎ and 𝑦 = 𝑌 + 𝑘 ❖ If 𝑒 > 1 then conic is hyperbola.
⇒𝑌 =𝑦−ℎ ❖ If 𝑒 = √2 then conic is rectangular
❖ If axis are rotated through an angle 𝜃 then the hyperbola.
coordinates of 𝑃(𝑥, 𝑦) and 𝑃(𝑋, 𝑌) are related ❖ If 𝑒 = 0 then conic is circle.
as ❖ If 𝑒 = ∞ then conic is pair of straight line.
𝑋 = 𝑥𝑐𝑜𝑠𝜃 + 𝑦𝑠𝑖𝑛𝜃 … (𝑖) ➢ If r is the radius of circle and 𝐶 (ℎ, 𝑘 ) is the
𝑌 = 𝑦𝑐𝑜𝑠𝜃 − 𝑥𝑠𝑖𝑛𝜃 … (𝑖𝑖) centre of the circle then the equation of a
circle is (𝑥 − ℎ)2 +(𝑦 − 𝑘 )2 = 𝑟 2
Slope formula
𝑦2 −𝑦1 ➢ The general equation of a circle is
➢ 𝑚= 𝑥 2 + 𝑦 2 + 2𝑔𝑥 + 2𝑓𝑦 + 𝑐 = 0 where g, 𝑓 and
𝑥2 −𝑥1
➢ 𝑚 = 𝑡𝑎𝑛𝛼 c are constants. The centre of circle is
➢ If 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 is the equation of line 𝐶 (−𝑔, −𝑓 ) and radius is 𝑟 = √𝑔2 +𝑓 2 − 𝑐
𝑎 𝑐𝑜𝑓𝑓 𝑜𝑓 𝑥
then slope, 𝑚 = − 𝑏 = − 𝑐𝑜𝑓𝑓 𝑜𝑓 𝑦 ➢ When two circle touches each other externally
𝑑𝑦 then |𝐶1 𝐶2 | = 𝑟1 + 𝑟2
➢ If 𝑦 = 𝑓(𝑥) then at (𝑥1 , 𝑦1 ) = 𝑚
𝑑𝑥 ➢ When circles touches each other internally and
❖ If lines are || then 𝑚1 = 𝑚2 one circle lie inside the other then
❖ If lines are ⊥ then 𝑚1 ∙ 𝑚2 = −1 |𝐶1 𝐶2 |=|𝑟1 − 𝑟2 |.
Type of lines
❖ 𝑦 = 𝑚𝑥 + 𝑐 (slope- intercept form)
Position of a point with respect to a circle i) If above result is positive then point 𝐴(𝑥1 , 𝑦1 )
Let 𝑥 2 + 𝑦 2 + 2𝑔𝑥 + 2𝑓𝑦 + 𝑐 = 0 be the lie outside the circle
equation of a circle and 𝐴(𝑥1 , 𝑦1 ) be a point ii) If this result is negative then point A lie in side
in the plane. the circle
we put the given point in the left hand side of If it is zero then point A lie on the circle
the equation of circle i.e.
𝑥12 +𝑦12 +2𝑔𝑥1 +2𝑓𝑦1 + 𝑐.
Table for parabola in standard form
Eq. of directrix Eq. of latus Eq. of axis of Focal
Equation & fig vertex Focus
line rectum line parabola distance
𝑦

𝑥
𝑂 (0, 0) 𝐹 (𝑎, 0) 𝑥 = −𝑎 𝑥=𝑎 𝑦=0 𝑥+𝑎

𝑦 2 = 4𝑎𝑥

𝑂 𝑥 (0, 0) 𝐹 (−𝑎, 0) 𝑥=𝑎 𝑥 = −𝑎 𝑦=0 𝑥−𝑎

𝑦 2 = −4𝑎𝑥

𝑥
(0, 0) 𝐹 (0, 𝑎) 𝑦 = −𝑎 𝑦=𝑎 𝑥=0 𝑦+𝑎
𝑂

𝑥 2 = 4𝑎𝑦
𝑦

𝑂
𝑥 (0, 0) 𝐹 (0, −𝑎) 𝑦=𝑎 𝑦 = −𝑎 𝑥=0 𝑦−𝑎

𝑥 2 = −4𝑎𝑦
Note:
For any parabola
i) The distance between vertex and focus of parabola is 𝑎
ii) The distance between vertex and directrix line is 𝑎
iii) The distance between foucs and directrix line is 2𝑎
iv) The length of lactus rectum is 4𝑎
 Table of ellipse in standard form
Eq. of Eq. of
Eq. of
Eq. & figure centre vertices Foci Co vertices major minor
latra recta
𝑎𝑥𝑖𝑠 𝑎𝑥𝑖𝑠
𝑦 ′ ′ ′
(0, 0) 𝐴(𝑎, 0) , 𝐴 (−𝑎, 0) 𝐹 (𝑐, 0) , 𝐹 (−𝑐, 0) 𝐵(0, 𝑏) , 𝐵 (0, −𝑏) Major minor 𝑥 = ±𝑐
i.e. (±𝑎 , 0) i.e. (±𝑐 , 0) i.e. (0, ±𝑏) axis axis or
𝑥 along along 𝑥 = ±𝑎𝑒
𝑂 x-axis y-axis
and and
2 2
𝑥 𝑦 eq. is eq. is
+ 2=1
𝑎 2 𝑏 𝑦=0 𝑥=0
′ ′ ′
(0, 0) 𝐴(0, 𝑎) , 𝐴 (0, −𝑎) 𝐹 (0, 𝑐 ) , 𝐹 (0, −𝑐 ) 𝐵(𝑏, 0) , 𝐵 ( −𝑏, 0) major
𝑦 minor 𝑦 = ±𝑐
i.e. (0, ±𝑎) i.e. (0, ±𝑐 ) i.e. (±𝑏, 0) axis axis or
𝑥 along along 𝑦 = ±𝑎𝑒
𝑂 y-axis x-axis
and and
eq. is eq. is
𝑦2 𝑥2 𝑥=0 𝑦=0
𝑎 2 + 2 = 1
𝑏

Note:
i) Distance b/w vertices=length of major axis = 2𝑎 ii) Distance b/w covertices = length of minor axis = 2𝑏
iii) Distance b/w foci = 2𝑐 = 2𝑎𝑒 iv) Sum of focal radii = 2𝑎 i.e |𝑃𝐹 | + |𝑃𝐹 ′ | = 2𝑎
2𝑎 2𝑏2
v) The distance between directrices line is vi) The length of latus rectum is
𝑒 𝑎
vii) In ellipse the eccentricity can be obtained as viii) In ellipse 𝑎 = 𝑏 + 𝑐 2
2 2

𝑐 𝑎2 −𝑏2
𝑒 = 𝑎 or 𝑒 = √ 𝑎2
Table of hyperbola in standard form
Eq. of
Eq. of Eq. of
Eq. & figure centre vertices Foci conjugate
transverse𝑎𝑥𝑖𝑠 latra recta
𝑎𝑥𝑖𝑠
𝑦 (0, 0) 𝐴(𝑎, 0) , 𝐴′ (−𝑎, 0) 𝐹 (𝑐, 0) , 𝐹 ′ (−𝑐, 0) Transverse conjugate 𝑥 = ±𝑐
i.e. (±𝑎 , 0) i.e. (±𝑐 , 0) axis x-axis its 𝑎𝑥𝑖𝑠 is along or
𝑥
eq. is 𝑦 = 0 y-asix and its 𝑥 = ±𝑎𝑒
𝑂
eq. is 𝑥 = 0

𝑥2 𝑦2
− =1
𝑎2 𝑏 2
(0, 0) 𝐴(0, 𝑎) , 𝐴′ (0, −𝑎) 𝐹 (0, 𝑐 ) , 𝐹 ′ (0, −𝑐 ) Transverse conjugate 𝑦 = ±𝑐
𝑦
i.e. ( 0, ±𝑎 ) (
i.e. 0, ±𝑐 ) axis y-axis its 𝑎𝑥𝑖𝑠 is along or
𝑥 eq. is 𝑥 = 0 x-asix and its 𝑦 = ±𝑎𝑒
𝑂
eq. is 𝑦 = 0

𝑦2 𝑥2
− =1
𝑎2 𝑏 2
Note:
Hyperbola has a central symmetry so it is called central conic.
Note:
ix)Distance b/w vertices=length of transverse axis = 2𝑎
x) Distance b/w foci = 2𝑐
xi)Differnce focal radii= 2𝑎 i.e |𝑃𝐹 ′ | − |𝑃𝐹 | = 2𝑎
2𝑎
xii) The distance between directrices line is
𝑒
2𝑏 2
xiii) The length of latus rectum is
𝑎
𝑐 𝑎2 +𝑏2
xiv) In hyperbola the eccentricity can be obtained as 𝑒 = or 𝑒=√
𝑎 𝑎2
xv) In hyperbola 𝑐 2 = 𝑎2 + 𝑏2

❖ The dot product of same vector is the

Chapter #7 square of magnitude of that vector i.e. 𝑣 ∙
𝑣 = |𝑣 |2
❖ Scalar product is commutative i.e. for any
❖ If 𝐴(𝑥1 , 𝑦1 ) & 𝐵(𝑥2 , 𝑦2 ) are two points two vectors 𝑢⃗ and 𝑣 we have 𝑢 ⃗ ∙ 𝑣 = 𝑣. 𝑢
⃗.
then vector from 2 points. ❖ Scalar product of two perpendicular
⃗⃗⃗⃗⃗
𝐴𝐵 = (𝑥2 −𝑥1 )𝑖 + (𝑦2 −𝑦1)𝑗 vectors is zero, i.e. if 𝑎 and 𝑏⃗ are two
perpendicular vectors, then 𝑎 ∙ 𝑏⃗ = 0.
❖ If v = xiˆ + y ˆj + zkˆ then
However, if 𝑎 ∙ 𝑏⃗ = 0 ⇒ either 𝑎 = 𝑜 or
|𝑣| = √𝑥 2 + 𝑦 2 + 𝑧 2
 𝑏⃗ = 𝑜 or 𝑎 ⊥ 𝑏⃗.
v vector
❖ Unit vector vˆ =  = ⃗𝑢∙𝑣
v magnitude of vector ❖ Projection of 𝑣 along 𝑢 ⃗ =
|𝑢⃗|
❖ Ratio formula ⃗𝑢
⃗ ∙𝑣
⃗⃗
⃗ +𝑘2 𝑎⃗
𝑘1 𝑏 ❖ Projection of 𝑢
⃗ along 𝑣 = |𝑣
⃗|
𝑟=
𝑘1 +𝑘2
Cross product of two vectors
❖ If 𝑢 ⃗ = 𝑥1 𝑖̂ + 𝑦1 𝑗̂ + 𝑧1 𝑘̂ and
𝑶
𝑣 = 𝑥2 𝑖̂ + 𝑦2 𝑗̂ + 𝑧2 𝑘̂ then
❖ Direction Cosine 𝑖̂ 𝑗̂ 𝑘̂
⃗ × 𝑣 = |𝑥1 𝑦1 𝑧1 |
𝑢
If v = xiˆ + y ˆj + zkˆ then direction cosines
𝑥 𝑦 𝑧 𝑥2 𝑦2 𝑧2
are 𝑐𝑜𝑠𝛼 = , 𝑐𝑜𝑠𝛽 = , 𝑐𝑜𝑠𝛾 = i) 𝑖̂ × 𝑖̂ = 𝑗̂ × 𝑗̂ = 𝑘̂ × 𝑘̂ = 0⃗ and 𝑖̂ × 𝑗̂ = 𝑘̂
|𝑣| |𝑣| |𝑣|
❖ If 𝛼 , 𝛽 , 𝛾 are direction angles then 𝑗̂ × 𝑘̂ = 𝑖̂ and 𝑘̂ × 𝑖̂ = 𝑗̂.
𝑐𝑜𝑠 2 𝛼 + 𝑐𝑜𝑠 2𝛽 + 𝑐𝑜𝑠 2𝛾 = 1 Also 𝑗̂ × 𝑖̂ = −𝑘̂ , 𝑘̂ × 𝑗̂ = −𝑖̂ and 𝑖̂ × 𝑘̂ = 𝑗̂.
Dot product properties ii) 𝑣 ∙ 𝑣 = ⃗0.
❖ 𝑢 ⃗ ∙ 𝑣 = |𝑢
⃗ ||𝑣|𝑐𝑜𝑠𝜃 ⃗ × 𝑣 = − (𝑣 × 𝑢
iii) 𝑢 ⃗)
 iv) The cross product of two parallel vectors is
❖ If u = x1iˆ + y1 ˆj + z1kˆ and
 zero i.e. if 𝑎 and 𝑏⃗ are two parallel vector
v = x2iˆ + y2 ˆj + z 2 kˆ then ⃗ = 0. However if 𝑎 × 𝑏⃗ = 0
then 𝑎 × 𝑏
𝑢
⃗ ∙ 𝑣 = 𝑥1 𝑥2 + 𝑦1 𝑦2 + 𝑧1 𝑧2
then either 𝑎 = ⃗0 or 𝑏⃗ = ⃗0 or 𝑎, 𝑏⃗ are
❖ iˆ.iˆ = ˆj. ˆj = kˆ.kˆ = 1
parallel
❖ iˆ. ˆj = ˆj.kˆ = kˆ.iˆ = 0
❖ Area of parallelogram = |𝑢
⃗ × 𝑣|
1
❖ Area of triangle = |𝑢
⃗ × 𝑣|
2
Scalar triple product Work done by a force
❖ If 𝑢 ⃗ = 𝑢1 𝑖̂ + 𝑢2 𝑗̂ + 𝑢3 𝑘̂ , ⃗⃗⃗⃗⃗
𝑊 = 𝐹 ⋅ 𝑑 where 𝑑 = 𝐴𝐵
𝑣 = 𝑣1 𝑖̂ + 𝑣2 𝑗̂ + 𝑣3 𝑘̂ and
⃗⃗ = 𝑤1 𝑖̂ + 𝑤2 𝑗̂ + 𝑤3 𝑘̂ then
𝑤 𝐹
𝑢1 𝑢2 𝑢3
⃗⃗ ) = | 𝑣1 𝑣2 𝑣3 |
⃗ . (𝑣 × 𝑤
𝑢
𝑤1 𝑤2 𝑤3
A 𝐵
⃗⃗⃗⃗⃗ = 𝑑
𝐴𝐵
Volume of parallelepiped = 𝑢 ⃗ . (𝑣 × 𝑤
⃗⃗ )
1 Moment of a force about a point = 𝑟 × 𝐹
⃗ . (𝑣 × 𝑤
Volume of tetrahedron = 𝑢 ⃗)
6