Coordinate systems

Welcome to Coordinate Systems!

In your previous studies, you’ve mastered straight lines and circles. Now, it’s time to level up! In this chapter of Further Pure Mathematics 1 (FP1), we explore two fascinating curves: the parabola and the rectangular hyperbola. These aren't just random shapes; they are "conic sections" that appear everywhere from the path of a kicked football to the design of satellite dishes.

Don't worry if these look a bit "maths-heavy" at first. We are going to break them down into simple pieces, focusing on how to describe them using both Cartesian and parametric equations.

1. The Parabola

You might recognize the parabola from quadratic graphs, but in FP1, we usually look at them lying on their side. The standard form we use is:

Cartesian Equation: \(y^2 = 4ax\)

Here, \(a\) is a constant that determines how "wide" or "narrow" the parabola opens.

Parametric Equations

Sometimes, it’s easier to describe a point on the curve using a third variable, \(t\), called a parameter. Think of \(t\) like a "time" stamp that tells you exactly where you are on the curve. For a parabola:

\(x = at^2\)
\(y = 2at\)

Any point on this parabola can be written as \((at^2, 2at)\). This is called the general point. It is a huge time-saver in exams!

Did you know? Parabolas have a unique reflective property. This is why satellite dishes and car headlights are parabolic—they reflect all incoming signals to a single point!

Key Takeaway: A parabola \(y^2 = 4ax\) can be represented by a single point \((at^2, 2at)\). If you see a coordinate like \((3t^2, 6t)\), you know immediately that \(a = 3\).

2. The Focus-Directrix Property

Every parabola has a "secret" point called the Focus and a "secret" line called the Directrix. The parabola is actually defined by these two things.

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

The Definition: Any point \(P\) on the parabola is exactly the same distance from the Focus as it is from the Directrix.
Mathematically: Distance \(PS\) = Distance \(PN\) (where \(N\) is the point on the directrix closest to \(P\)).

Quick Review:
For \(y^2 = 4ax\):
1. Focus is \((a, 0)\).
2. Directrix is \(x = -a\).
3. Vertex (the tip) is at \((0, 0)\).

3. The Rectangular Hyperbola

Next up is the rectangular hyperbola. You’ve seen this before as the reciprocal graph \(y = 1/x\), but we use a more general version in FP1.

Cartesian Equation: \(xy = c^2\)

This curve has two separate parts (branches) and never touches the \(x\) or \(y\) axes. These axes are called asymptotes.

Parametric Equations

Just like the parabola, we can use a parameter \(t\) to find any point on the hyperbola:

\(x = ct\)
\(y = \frac{c}{t}\)

The general point is \((ct, \frac{c}{t})\).

Memory Aid: For the hyperbola, notice that if you multiply the \(x\) and \(y\) coordinates together: \(ct \times \frac{c}{t} = c^2\). This confirms the Cartesian equation \(xy = c^2\)!

Key Takeaway: For a hyperbola \(xy = c^2\), the general point is \((ct, c/t)\). If \(xy = 25\), then \(c = 5\), and the general point is \((5t, 5/t)\).

4. Tangents and Normals

This is where your differentiation skills from P1 come in handy! You will often be asked to find the equation of a tangent (a line just touching the curve) or a normal (a line perpendicular to the tangent) at a specific point.

Step-by-Step: Finding the Gradient

To find the gradient of the tangent, you need \(\frac{dy}{dx}\). Since we often use Cartesian equations here, we rearrange them first:

For the Parabola:
Rearrange \(y^2 = 4ax\) to \(y = \sqrt{4ax} = 2\sqrt{a}x^{1/2}\).
Differentiate: \(\frac{dy}{dx} = 2\sqrt{a}(\frac{1}{2}x^{-1/2}) = \frac{\sqrt{a}}{\sqrt{x}}\).

For the Hyperbola:
Rearrange \(xy = c^2\) to \(y = c^2x^{-1}\).
Differentiate: \(\frac{dy}{dx} = -c^2x^{-2} = -\frac{c^2}{x^2}\).

Common Mistake to Avoid

Students often forget that the normal is perpendicular to the tangent.
If the gradient of the tangent is \(m\), the gradient of the normal is \(-\frac{1}{m}\). Always double-check which one the question is asking for!

Finding the Equation of the Line

Once you have the gradient (\(m\)) and a point \((x_1, y_1)\), use the formula you know from P1:
\(y - y_1 = m(x - x_1)\)

Key Takeaway Summary:
1. Parabola: \(y^2 = 4ax\); point \((at^2, 2at)\); Focus \((a, 0)\); Directrix \(x = -a\).
2. Hyperbola: \(xy = c^2\); point \((ct, c/t)\).
3. Tangents: Use \(\frac{dy}{dx}\) to find the gradient.
4. Normals: Gradient is \(-\frac{1}{\text{tangent gradient}}\).

Final Encouragement

Coordinate systems can feel abstract at first because of all the \(a\)'s, \(c\)'s, and \(t\)'s. Try replacing them with numbers in your head to see how the equations work. With a bit of practice identifying the "general point," you'll find these curves much easier to handle than they look!