Parabolas and hyperbolas (FP1) are two of the curves got by slicing a double-cone. They are also two of the curves got by writing second-degree equations in x and y. The other two are ellipses and hyperbolas.

In FP1 we described parabolas either by the Cartesian equation y^{2} = 4ax or by the “parametric form” of the typical point (at^{2}, 2at)

Now we describe an ellipse either by the Cartesian equation

Or by the parametric form

With the parabola, we called t “timestamp”. What does it represent here?

What happens when a = b?

Do Ex.2A Q.1 and 2

**Tangents and normals to ellipses**
How did you find the equation of the tangent at (ap^{2}, 2ap) to the parabola with typical point (at^{2}, 2at)?

And the normal?

Adapt that method for ellipse

Do Ex.2B Q.1-5

**Hyperbolas**
Recall parametric form of rectangular hyperbolas in FP1

Now we study not-necessarily-rectangular hyperbolas

Draw these diagrams. What are x and y in terms of α and t?

What are the asymptotes of these hyperbolas?

Graphics showing parametric forms of hyperbola

(Note that the cosh-sinh parametric form gives only the right half of the hyperbola).

Four ways to find the asymptotes

1. Look in the formula book: asymptotes are y = ± (^{b}⁄_{a}) x

2. When x and y get very big, ^{x2}⁄_{a2} − ^{y2}⁄_{b2} = 1

is almost the same as ^{x2}⁄_{a2} − ^{y2}⁄_{b2} = 0

i.e. as (^{x}⁄_{a} + ^{y}⁄_{b})(^{x}⁄_{a} − ^{y}⁄_{b}) = 0

so ^{x}⁄_{a} = ± ^{y}⁄_{b} are the asymptotesz.

3. When θ approaches π/2 or 3π/2, then ^{y}⁄_{x}, which is (^{b sin θ}⁄_{a}), approaches (^{b}⁄_{a})

4. When t approaches ∞, then sinh t is almost the same as ± cosh t, so y = ± (^{b}⁄_{a}) x

Do Ex.2C Q.2

**Tangents and normals to hyperbolas using both parametric forms**

Do Ex.2D Q.1-2

Homework: Complete Ex.2A Q.1-2, Ex.2B Q.1-5, Ex.2C Q.2, Ex.2D Q.1-2

**Focus, directrix, and eccentricity**
Draw these ellipses. If an ellipse is more eccentric, then its two foci are….?

Definitions and algorithms

Ex.2E Q.1-4

Ex.2E Q.5-6

**Loci**

Ex.2F Q.2,6

Homework: complete Ex.2D Q.1-2, Ex.2E Q.1-6, Ex.2F Q.2,6

**Problems**
Ex.2G Q.1

FP3 past paper questions: conics

Three ways of proving that the normal at P bisects angle FPF’, if F and F’ are the foci of an ellipse

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