Welcome to the World of Curves!

In your previous math journeys (P1 and P2), you’ve spent a lot of time with straight lines and circles. In Further Pure 1 (FP1), we are going to branch out and look at two very special types of curves: the Parabola and the Rectangular Hyperbola. These aren't just random shapes; they are the paths planets take, the shapes of satellite dishes, and even the curves of cooling towers!

Don't worry if this seems a bit abstract at first. We are going to break it down piece by piece. By the end of these notes, you’ll see that these curves follow very logical rules.

1. Prerequisite Check: What You Need to Know

Before we dive in, make sure you are comfortable with:

  • Coordinates: Plotting points \((x, y)\).
  • Differentiation: Finding the gradient function \(dy/dx\).
  • Equation of a Line: Using \(y - y_1 = m(x - x_1)\).
  • Reciprocal Graphs: Knowing that \(y = 1/x\) creates a curve that never touches the axes.

2. The Parabola: The "Perfect" Reflection

A Parabola is the U-shaped curve you see when you throw a ball in the air. In FP1, we usually look at parabolas that open to the right.

The Cartesian Equation

The standard equation for a parabola is:
\(y^2 = 4ax\)

Here, \(a\) is a constant that determines how "wide" or "narrow" the parabola is. The point \((0,0)\) is the vertex (the turning point).

The Parametric Equation

Sometimes, it's easier to describe \(x\) and \(y\) using a third variable, \(t\), called a parameter. Think of \(t\) like "time"—it tells you where you are on the curve at a specific moment.
For the parabola \(y^2 = 4ax\), the parametric equations are:
\(x = at^2\)
\(y = 2at\)

The Focus-Directrix Property

This is the "secret definition" of a parabola. Every parabola has a special point called the Focus and a special line called the Directrix.

  • Focus (S): Located at \((a, 0)\).
  • Directrix: The vertical line with the equation \(x = -a\).

The Property: Any point on the parabola is exactly the same distance from the Focus as it is from the Directrix.
Analogy: Imagine standing in a field. You are a point on the parabola if you are exactly 10 meters away from a specific tree (the focus) AND 10 meters away from a long fence (the directrix).

Quick Review Box:
For \(y^2 = 4ax\):
- Focus is \((a, 0)\)
- Directrix is \(x = -a\)
- General point is \((at^2, 2at)\)

3. The Rectangular Hyperbola: The Reciprocal Classic

You’ve seen \(y = 1/x\) before. A Rectangular Hyperbola is just a slightly more general version of that.

The Cartesian Equation

The equation is:
\(xy = c^2\) (which is the same as \(y = \frac{c^2}{x}\))

In this curve, the \(x\)-axis and \(y\)-axis are asymptotes. The curve gets closer and closer to them but never actually touches them.

The Parametric Equation

Just like the parabola, we can use a parameter \(t\):
\(x = ct\)
\(y = \frac{c}{t}\)

Any point on this hyperbola can be written as \((ct, \frac{c}{t})\).

Did you know?
It’s called "rectangular" because its asymptotes (the lines it never touches) meet at a 90-degree angle, just like the corner of a rectangle!

Key Takeaway:
The parabola is defined by its Focus and Directrix, while the hyperbola is defined by its constant \(c\) and its relationship with the axes.

4. Tangents and Normals

This is where we use our calculus skills to find the equation of a line that just touches the curve (the Tangent) or the line that is perpendicular to it (the Normal).

Step-by-Step: Finding the Gradient

To find the equation of a tangent or normal at a specific point, you first need the gradient \(m\). We do this by differentiating the Cartesian equation.

For the Parabola:

1. Start with \(y^2 = 4ax\).
2. Rewrite it as \(y = \sqrt{4ax} = 2a^{1/2}x^{1/2}\).
3. Differentiate: \( \frac{dy}{dx} = 2a^{1/2} \cdot \frac{1}{2}x^{-1/2} = \frac{\sqrt{a}}{\sqrt{x}} \).
4. Plug in your \(x\)-coordinate to find the gradient \(m\).

For the Hyperbola:

1. Start with \(y = \frac{c^2}{x} = c^2x^{-1}\).
2. Differentiate: \( \frac{dy}{dx} = -c^2x^{-2} = -\frac{c^2}{x^2} \).
3. Plug in your \(x\)-coordinate to find the gradient \(m\).

Finding the Final Equation

Once you have the gradient \(m\) and the point \((x_1, y_1)\):
- Tangent: Use \(y - y_1 = m(x - x_1)\).
- Normal: Use the perpendicular gradient \(-\frac{1}{m}\) in the same formula: \(y - y_1 = -\frac{1}{m}(x - x_1)\).

Common Mistake to Avoid:
When finding the Normal gradient, don't forget to flip and change the sign of the tangent gradient. If the tangent gradient is \(1/2\), the normal gradient is \(-2\)!

5. Summary and Memory Aids

If you're feeling a bit overwhelmed, try these simple memory tricks:

  • Parabola Pair: In the parametric form \( (at^2, 2at) \), the \(2\) is in the \(y\) term, just like the \(2\) is in the power of \(y^2\).
  • Hyperbola Help: In \(xy = c^2\), \(x\) and \(y\) are "multiplied together." In the parametrics \(ct\) and \(c/t\), if you multiply them together, the \(t\) cancels out to give you \(c^2\)!

Final Key Takeaways:
1. Parabola: \(y^2 = 4ax\). Focus at \((a,0)\), Directrix at \(x = -a\).
2. Hyperbola: \(xy = c^2\). Parametrics are \(x=ct, y=c/t\).
3. Calculus: Use differentiation on the Cartesian form to find gradients for tangents and normals.

You've got this! These curves are just patterns. Practice finding the focus and directrix a few times, and it will become second nature!