ParametricsSummary Parametric equation of curves Eliminate the parameter Chords of a parabola Parametric equation Find the gradient Use the point-gradient form Focal chord Sub the coordinates of the focus into the equation of the chord. Tangents and normals Parametric approach Differentiate to find the gradient Use the point-gradient form to find the equation Cartesian approach Differentiate to find the gradient Use the point-gradient form to find the equation Chord of contact (no proof necessary)

Parametric equationsIt is a common practice in mathematics to express two related variables, say and , in terms of a third variable, say or , so that

These equations are called parametric equations and or is called the parameter.Example 1Find the Cartesian equation of the curve whose parametric equations are

Rearranging the first equation, we get , and since , it follows that .Example 2Find the Cartesian equation of the curve whose parametric equations are

ExercisesFind the Cartesian equation of the curves whose parametric equations are:1. 2. 3. 4.

Parametric equation of the ParabolaThe parabola can be represented by the parametric equations:

The point is a variable point of the parabola depending on the value of .Chords of a parabolaSuppose that and are two distinct points on the parabola .Gradient of chord:

Equation of chord:

(Note that if parameters and are exchanged, formulae for the gradient and the equation remain the same.This is because the chord is the same line as the chord .)Focal chordA focal chord is a chord that passes through the focus of a parabola. Substituting into the equation of the chord above, is a focal chord if and only if .

Tangents and normalsSuppose is a point on the parabola with equation The gradient of the tangent at is .

Parametric equation of the tangent

The tangent at is Tangents from an external pointa) Differentiating,Equation of tangent:

b) Substutiting ,

Points of contact are The corresponding tangents are

Intersection of two tangents(Solving simultaneous equations)

Thus, the tangents at and meet at .Parametric equation of the normalGradient of normalEquation of normal:

Cartesian equation of the tangentSuppose is any point on the parabola

Answers to exercises1. 2. 3. 4.