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

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Ellipses[edit]

Tangent to an ellipse: Cartesian approach[edit]

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

Hyperbolae[edit]

Tangent to a hyperbola: Cartesian approach[edit]

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 approach[edit]

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