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

Ellipses Tangent to an ellipse: Cartesian approach The Cartesian equation of the ellipse is . Differentiating (using the technique of Implicit differentiation to simplify the process) to find the gradient: Hyperbolae Tangent to a hyperbola: Cartesian approach 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 The gradient of the normal is given by , i.e., . Finding the equation,