FP3 conics (ellipses and hyperbolas)

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 y2 = 4ax or by the “parametric form” of the typical point (at2, 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 (ap2, 2ap) to the parabola with typical point (at2, 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 = ± (ba) x

2. When x and y get very big, x2a2y2b2 = 1

is almost the same as x2a2y2b2 = 0

i.e. as (xa + yb)(xayb) = 0

so xa = ± yb are the asymptotesz.

3. When θ approaches π/2 or 3π/2, then yx, which is (b sin θa), approaches (ba)

4. When t approaches ∞, then sinh t is almost the same as ± cosh t, so y = ± (ba) 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