Having learnt about the prime conic 'The Parabola', it is of very much importance to know about the two remaining conics namely 'The Ellipse' and 'The Hyperbola'. They both possess similar features but differ in terms of eccentricity from each other. The comparison of both of them is very interesting.

* Given the equations of the ellipse and hyperbola in their standard forms, nine items such as 'eCSAZ BELT' can be written easily.

     C - Centre (o, o)
     S - Focii (± ae, o)
     A - vertices (± a,o)
     Z - directrices - x = ± 
     B - points (o, ± b)
     E - Equations of latera recta: x = ± ae
     L - Length of Latus Rectum 

     T - Tangents at vertices: x = ± a
* Both are symmetrical about the coordinate axes.
* '2a' is the length of major axis in the case of ellipse (a > b) and the same is the length of transverse axis in the case of hyperbola.
* The sum of the focal distances of any point on an ellipse is constant and equal to the major axis.
* The focal distances of any point (x, y) on the ellipse are a ± ex.

*  is the tangent in slope form for ellipse and  is the tangent in slope form for hyperbola.
* Normal at (x₁,  y₁):  =  a² - b² (ellipse)   =  a² + b² (hyperbola)
* P (θ): (a cosθ, b sinθ) (ellipse) (a secθ, b tanθ) (hyperbola)
* Tangent at 'θ':  (ellipse)   (hyperbola)
* Normal at 'θ':  (ellipse)   (hyperbola)
* The two diameters y = mx and y = m₁x are said to be conjugate
    if mm1= -b²/a² for an ellipse and mm₁= b²/a² for hyperbola respectively.
* Director circle for ellipse is given by the equation x² + y²  =  a² + b² and the same is given for hyperbola by x² + y² = a² - b².
* Auxiliary circle remains the same as x² + y² = a² for both of the conics.
* Chord joining 'θ₁', and 'θ₂':

* x cosα + y sinα = p will become a tangent if
        a² cos² α + b² sin² α =  p² (for ellipse) 
        a² cos² α - b² sin² α  =  p² (for hyperbola)
* Condition for conjugate lines:
        a l₁l₂ + b m₁m₂ =  n₁n₂ (ellipse)
        a l₁l₂ - b m₁m₂ = n₁n₂ (hyperbola)
* Equation of hyperbola: 
     Equation of conjugate hyperbola: .
* Hyperbola is a significant conic by virtue of it possessing asymptotes.
* An asymptotes is a straight line which meets a hyperbola in two co-incident points at infinity, but itself does not altogether lie at infinity.

* y = ±  x are said to represent the a asymptotes. And hence these are considered as straight lines passing through the centre of the Hyperbola.
* The joint equation of the asymptotes is 
* If the asymptotes are at right angles to each other, then rectangular hyperbola is formed.
* For a rectangular hyperbola, the transverse and conjugate axes are equal and hence x² - y² =  a² will be its equation.
* A rectangular hyperbola is also termed as equilateral hyperbola.
*  is the eccentricity of a rectangularhyperbola.
* Two diameters of an ellipse (a hyperbola) are said to be conjugate if each bisects chords parallel to the other.
* When the two conjugate diameters are equal in length, they are called equi-conjugate diameters.
* The equi-conjugate diameters are obviously the diagonals of the rectangle formed by tangents at the ends of major and minor axes.

* When an ellipse slides between two straight lines which are at right angles to one another, then the locus of its centre is a circle.