Property Circle Parabola Ellipse Hyperbola Rect. Hyp.

Eccentricity - e=1 e<1 e>1 𝑒= 2

General 𝑦 = ±4𝑎𝑥 (ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙)
𝑥 + 𝑦 = 𝑎 + =1 - =1 𝑥𝑦 = 𝑐
Equation 𝑥 = ±4𝑎𝑦 (𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙)
𝑐
Parametric Form 𝑎 cos 𝜃 , 𝑎 sin 𝜃) (±𝑎𝑡 , 2𝑎𝑡) (a cos 𝜃, 𝑏 sin 𝜃) (𝑎 sec 𝜃, 𝑏 tan 𝜃) (𝑐𝑡, )
𝑡
Diametric 𝑥−𝑥 𝑥−𝑥
+ 𝑦−𝑦 𝑦−𝑦 =0 - - - -
Equation
General
𝑥 + 𝑦 + 2𝑔𝑥 + 2𝑓𝑦 + 𝑐 = 0 - - - = −1 𝑥𝑦 = −𝑐
Equation
Standard
(𝑥 − ℎ) + (𝑦 − 𝑘) = 𝑎 (𝑦 − 𝑘) = ±4𝑎(𝑥 − ℎ) - - -
Equation
Radius = a 𝑥𝑦 = 𝑐 :
If a>b:
Centre = (-g, -f) (General) Same as ellipse: TA: x-y=0
Semimajor axis = a CA: x+y=0
Centre = (h, k) (Standard) Major axis is
Information Vertex = (0, 0) Semiminor axis = b 𝑥𝑦 = −𝑐 :
Diametric: (𝑥 , 𝑦 ) and Transverse axis, TA: x+y=0
Provided Focus = a If b>a:
𝑥 , 𝑦 are ends of diameter Minor axis is CA: x-y=0
Semimajor axis = b
Conjugate Axis Vertices:
Semiminor axis = a
(±𝑐, ±𝑐)
- 𝑎 𝑎
Directrices 𝑥 = −𝑎 𝑥= ± 𝑥= ± 𝑥 + 𝑦 = ± 2𝑐
𝑒 𝑒
Condition |𝑎ℎ + 𝑏𝑘 + 𝑐| 𝑐= (horizontal)
=𝑟 𝑐 =𝑎 𝑚 +𝑏 𝑐 =𝑎 𝑚 −𝑏 Same
for Tangent 𝑎 + 𝑏 c = −a𝑚 (𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙)
𝑥𝑦 + 𝑦𝑥
Tangent 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦
T=0 T=0 + =1 − =1 2
(point form) 𝑎 𝑏 𝑎 𝑏 −𝑐 = 0
Tangent 𝑦 = 𝑚𝑥 + (horizontal)
- 𝑦 = 𝑚𝑥 ± 𝑎 𝑚 +𝑏 𝑦 = 𝑚𝑥 ± 𝑎 𝑚 −𝑏
(slope form) 𝑦 = 𝑚𝑥 − 𝑎𝑚 (vertical)
Tangent 𝑦𝑡 = 𝑥 + 𝑎𝑡 (horizontal) 𝑥 𝑦 𝑥 𝑦 1
- 𝑥𝑡 = 𝑦 + 𝑎𝑡 (vertical) cos 𝜃 + sin 𝜃 = 1 sec 𝜃 − tan 𝜃 = 1 𝑚=−
(Parametric) 𝑎 𝑏 𝑎 𝑏 𝑡
Angle b/w 2 ℎ − 𝑎𝑏 2 ℎ − 𝑎𝑏 2𝑎𝑏 𝑆 2𝑎𝑏 −𝑆
tan tan tan tan Same
Tangents |𝑎 − 𝑏| |𝑎 − 𝑏| 𝑥 + 𝑦 − 𝑎 −𝑏 𝑥 +𝑦 −𝑎 +𝑏
Point of
Intersection - (𝑎𝑡 𝑡 , 𝑎 𝑡 + 𝑡 ) - - -
of Tangents
x = -a
Director Circle 𝑥 + 𝑦 = 2𝑎 𝑥 +𝑦 =𝑎 +𝑏 𝑥 +𝑦 = 𝑎 −𝑏 Same
y = -a
Condition 𝑎 −𝑏 𝑚 𝑎 +𝑏 𝑚
Passes through centre - 𝑐= ± 𝑐= ± Same
for Normal 𝑎 +𝑏 𝑚 𝑎 −𝑏 𝑚
Normal −𝑦 𝑎 𝑥 𝑏 𝑦 𝑎 𝑥 𝑏 𝑦
Passes through centre 𝑦−𝑦 = (𝑥 − 𝑥 ) − =𝑎 −𝑏 + =𝑎 +𝑏
(Point) 2𝑎 𝑥 𝑦 𝑥 𝑦
Normal 𝑎 −𝑏 𝑚 𝑎 +𝑏 𝑚
Perpendicular to tangent 𝑦 = 𝑚𝑥 − 2𝑎𝑚 − 𝑎𝑚 𝑦 = 𝑚𝑥 ± 𝑦 = 𝑚𝑥 ±
(Slope) 𝑎 +𝑏 𝑚 𝑎 −𝑏 𝑚
Normal 𝑎𝑥 𝑏𝑦 𝑎𝑥 𝑏𝑦
- 𝑦 + 𝑥𝑡 = 2𝑎𝑡 + 𝑎𝑡 − =𝑎 −𝑏 + =𝑎 +𝑏
(Parametric) cos 𝜃 sin 𝜃 sec 𝜃 tan 𝜃
Angle b/w 2 ℎ − 𝑎𝑏
- tan - -
Normals |𝑎 − 𝑏|
2𝑏
𝑎 > 𝑏:
LLR - 4a 𝑎 2𝑎 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 2a = 2b
2𝑎 2𝑏 (𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙)
𝑎 < 𝑏:
𝑏
Equation of
chord with given T = S1 T = S1 T = S1 T = S1 Same
midpoint
Chord of Contact T = S1 T = S1 T = S1 T = S1 Same