Welcome to Further Coordinate Systems!

In your earlier studies, you explored the parabola and the rectangular hyperbola. In this chapter of FP3, we are going to expand our horizons to look at two more fascinating curves: the Ellipse and the Hyperbola. These shapes aren't just mathematical abstractions; they describe the paths of planets around the sun and the way cooling towers are built!

Don’t worry if these look a bit intimidating at first. We’ll break them down into bite-sized pieces, looking at their equations, their special geometric properties, and how to find lines that just touch them (tangents).


1. The Ellipse

Think of an ellipse as a "squashed circle." While a circle has one radius, an ellipse has two main "dimensions": the semi-major axis \(a\) and the semi-minor axis \(b\).

Cartesian and Parametric Equations

The standard Cartesian equation for an ellipse centered at the origin \((0,0)\) is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] Where \(a\) is the distance from the center to the furthest edge along the x-axis, and \(b\) is the distance to the edge along the y-axis.

Sometimes it’s easier to describe a point on the ellipse using a single angle, \(t\). These are the parametric equations:
\(x = a \cos t\)
\(y = b \sin t\)

Quick Review: To turn parametric equations back into Cartesian, remember your trig identity: \(\cos^2 t + \sin^2 t = 1\). Just rearrange for \(\cos t = \frac{x}{a}\) and \(\sin t = \frac{y}{b}\), then square and add!

Focus-Directrix Properties and Eccentricity

Every ellipse has two special points called foci (plural of focus) and two lines called directrices. The shape is defined by how "stretched" it is, which we measure using eccentricity (\(e\)).

  • For an ellipse, the eccentricity is always between 0 and 1 (\(0 < e < 1\)).
  • Important Formula: \(b^2 = a^2(1 - e^2)\). This is how you find \(e\) if you know \(a\) and \(b\).
  • Foci: Located at \((\pm ae, 0)\).
  • Directrices: The vertical lines with equations \(x = \pm \frac{a}{e}\).

Did you know? If you take any point on the ellipse, the ratio of its distance from a focus to its distance from the corresponding directrix is always equal to \(e\). This is the "Focus-Directrix property."

Key Takeaway: The ellipse is a closed loop defined by \(a, b,\) and an eccentricity \(e < 1\). Use the identity \(\cos^2 t + \sin^2 t = 1\) to switch between equation forms.


2. The Hyperbola

A hyperbola looks like two mirrored "infinite" bows. It is fundamentally different from an ellipse because the "arms" never meet; instead, they go off to infinity.

Cartesian and Parametric Equations

The Cartesian equation for a hyperbola is very similar to the ellipse, but with a minus sign:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] Notice that \(x^2\) is positive, meaning this hyperbola opens left and right.

There are two ways to write the parametric equations for a hyperbola:

  1. Using Trigonometry: \(x = a \sec t, y = b \tan t\) (Using the identity \(\sec^2 t - \tan^2 t = 1\))
  2. Using Hyperbolic Functions: \(x = a \cosh t, y = b \sinh t\) (Using the identity \(\cosh^2 t - \sinh^2 t = 1\))

Geometric Properties

Like the ellipse, the hyperbola has foci and directrices, but it is much "wilder," so its eccentricity is greater than 1.

  • Eccentricity (\(e\)): \(e > 1\).
  • Important Formula: \(b^2 = a^2(e^2 - 1)\). (Notice the order is swapped compared to the ellipse!)
  • Foci: Located at \((\pm ae, 0)\).
  • Directrices: Vertical lines \(x = \pm \frac{a}{e}\).

Common Mistake: Students often mix up the \(b^2\) formulas for ellipses and hyperbolas.
Memory Aid:
- Ellipse is Enclosed (small \(e\)), so use \((1 - e^2)\).
- Hyperbola is Hyper (large \(e\)), so use \((e^2 - 1)\).

Key Takeaway: The hyperbola has two branches and an eccentricity \(e > 1\). It can be represented using either \(\sec/\tan\) or \(\cosh/\sinh\) parameters.


3. Tangents and Normals

Finding the equation of a tangent (a line that just grazes the curve) or a normal (a line perpendicular to the tangent) is a core skill in FP3.

The Condition for Tangency

If you have a straight line \(y = mx + c\), how do you know if it's a tangent to our curves? There are specific "conditions" (shortcut formulas) you should know:

  • For the Ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\): The line is a tangent if \(c^2 = a^2m^2 + b^2\).
  • For the Hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\): The line is a tangent if \(c^2 = a^2m^2 - b^2\).

How to find the Equation of a Tangent

If you need to find the tangent at a specific point \((x_1, y_1)\), you can use differentiation. Since you are in FP3, you can use Implicit Differentiation:

  1. Differentiate the Cartesian equation with respect to \(x\). (e.g., for an ellipse: \(\frac{2x}{a^2} + \frac{2y}{b^2} \frac{dy}{dx} = 0\)).
  2. Rearrange to find the gradient \(\frac{dy}{dx}\).
  3. Use the point-slope formula: \(y - y_1 = m(x - x_1)\).

Encouragement: If the differentiation feels messy, take it slow! Remember that \(a^2\) and \(b^2\) are just constant numbers, not variables.

Key Takeaway: Use the \(c^2\) formulas for quick checks, or use differentiation to find the tangent at a specific point. The normal always has a gradient of \(-\frac{1}{m}\).


4. Loci Problems

A locus (plural: loci) is simply a set of points that satisfy a certain rule. In this chapter, you might be asked to find the path traced by the midpoint of a moving chord or the intersection of two shifting tangents.

Step-by-Step Approach to Loci:

  1. Identify the "Moving" Point: Call the coordinates of the point you are interested in \((X, Y)\).
  2. Use Parametric Coordinates: If the point depends on a point on an ellipse, start with \((a \cos t, b \sin t)\).
  3. Relate \(X\) and \(Y\) to the parameter: Write expressions for \(X\) and \(Y\) in terms of \(t\) (or whatever parameter is used).
  4. Eliminate the parameter: Use trig identities (like \(\sin^2 + \cos^2 = 1\)) to get an equation with only \(X, Y, a,\) and \(b\).

Analogy: Imagine a person walking around a circular track while a drone records their position. The "locus" is the line the drone draws on its map. Even if the person stops or changes speed (the parameter \(t\)), the shape of the path (the locus) stays the same.

Key Takeaway: Loci problems are all about "getting rid of the middleman." Use your trig or hyperbolic identities to eliminate the parameter and find the hidden Cartesian path.


Final Summary Table

Shape: Ellipse
Equation: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
Eccentricity: \(e < 1\)
Key Formula: \(b^2 = a^2(1 - e^2)\)

Shape: Hyperbola
Equation: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
Eccentricity: \(e > 1\)
Key Formula: \(b^2 = a^2(e^2 - 1)\)

You've got this! Mastery of this chapter comes from practicing the switch between parametric and Cartesian forms. Keep those trig identities handy!