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