Circle and Conics

Circle
Equation of Circle 

Standard Form

(x-h)2 + (y-k)2 = 0

Diameter Form

(x-x1)(x-x2) + (y-y1)(y-y2) = 0

General Form

x2 + y2 + 2gx + 2 fy + c = 0 (Note : There is no term of xy in this equation !)

Centre co-ordinates = (-g, -f)

Radius = sqrt [g2 + f2 - c]

Parametric Form

(x,y)= (asinθ + acosθ )

Equation of a circle passing through intersection of two other circles 

Tangents

The lengths of tangents from a point (x1,y1) is :

sqrt [x12 + y12 + 2gx1 + 2fy1 + c]

Normals

Intersection

Parabola
Equation

y2 = 4ax

Directrix

x = - a

Focus

(a,0)

Latus Rectum

LR = 4a

Parametric Form

x = at2 ; y = 2at

Tangents and Normals at (x1,y1)

Tangent : yy1= 2a (x + x1) ; t =  1/m

Normal : [x - x1] / [-2a] = [y - y1] / y1

Ellipse
Equation

x2/a2 + y2/b2 = 1

Directrix

Focus

(+ae,0) ; (-ae,0)

Axes

Minor Axis =         Major Axis =

Parametric Form

(x,y) = ( asinθ, bcosθ )

Tangents and Normals

Tangent : xx1 / a2 + yy1/ b2 = 1 ;  y = mx + sqrt (a2m2 - b2)

Normals : 

Hyperbola
Equation

x2/a2 - y2/b = 1

Directrix

Focus

Axes

Parametric Form

x = (asecθ, btanθ)

Tangents and Normals

Tangent : xx1 / a2 - yy1/ b2 = 1 ; y = mx - sqrt (a2m2 - b2)

Normals : 

Tips and Tricks
1) The properties of radius of circle are useful for solving circle based questions.