HSC Extension 1 and 2 Mathematics/4-Unit/Conics

EllipsesEdit

Tangent to an ellipse: Cartesian approachEdit

The Cartesian equation of the ellipse is  . Differentiating (using the technique of Implicit differentiation to simplify the process) to find the gradient:

HyperbolaeEdit

Tangent to a hyperbola: Cartesian approachEdit

The Cartesian equation of the hyperbola is  . Differentiating (using the technique of Implicit differentiation to simplify the process) to find the gradient:

 

We can then substitute this into our point-gradient form,  , using the point  :

at  ,  .
 
But we know that   from the definition of the hyperbola, so
 

Normal to a hyperbola: Cartesian approachEdit

The gradient of the normal is given by  , i.e.,  . Finding the equation,