Welcome to the World of Coordinate Systems!
In your standard A-Level Maths, you’ve spent a lot of time with straight lines and circles. Now, in Further Maths, we are going to explore some "curvier" and more exciting shapes: the Parabola and the Rectangular Hyperbola. These aren't just abstract drawings; they describe the path of a thrown ball, the shape of a satellite dish, and even how planets move! Don't worry if these look intimidating; we’ll break them down piece by piece.
1. The Parabola: \(y^2 = 4ax\)
You’ve seen parabolas like \(y = x^2\) before. In Further Pure 1, we usually turn them on their side. The standard Cartesian equation we use is \(y^2 = 4ax\).
What is \(a\)?
Think of \(a\) as the "DNA" of the parabola. It is a constant that tells us how wide or narrow the curve is. It also tells us exactly where the "special points" of the parabola are located.
Parametric Equations: The "t" Shortcut
Sometimes, using \(x\) and \(y\) in one equation is messy. Instead, we can describe every point on the parabola using a single helper variable called a parameter, usually written as \(t\).
For the parabola \(y^2 = 4ax\), any point \(P\) can be written as:
\(x = at^2\)
\(y = 2at\)
Example: If \(a = 3\), a point on the parabola could be represented by \((3t^2, 6t)\). If you plug these into \(y^2 = 4ax\), you'll see they always work!
Quick Review: Parabola Basics
- Cartesian form: \(y^2 = 4ax\)
- Parametric form: \((at^2, 2at)\)
- Vertex: The "tip" of the curve, which is always at \((0, 0)\) in this form.
Common Mistake: Don't mix up the \(x\) and \(y\) in the parametric form. Remember that the \(y\) value is the one with the 2, and the \(x\) value is the one where the \(t\) is squared!
2. The Focus and Directrix
Every parabola has a "magic" point called the Focus and a "magic" line called the Directrix.
The Definition
A parabola is actually defined by a rule: Every point on the curve is exactly the same distance from the Focus as it is from the Directrix.
- The Focus (\(S\)): A point located at \((a, 0)\).
- The Directrix (\(L\)): A vertical line with the equation \(x = -a\).
Analogy: Imagine the Focus is a campfire and the Directrix is a cold stone wall. To stay perfectly comfortable, you must walk along a path where you are always exactly as far from the heat as you are from the cold. That path you walk is a parabola!
Did you know? This property is why satellite dishes are parabolic. Any signal hitting the dish reflects perfectly into the Focus, where the receiver sits!
3. The Rectangular Hyperbola: \(xy = c^2\)
The next shape is the Rectangular Hyperbola. You might recognize this from GCSE as the "1/x graph." Its Cartesian equation is \(xy = c^2\) (where \(c\) is a constant).
Parametric Equations
Just like the parabola, we can use a parameter \(t\) for the hyperbola:
\(x = ct\)
\(y = \frac{c}{t}\)
Memory Aid: Notice that if you multiply \(x\) and \(y\) together (\(ct \times \frac{c}{t}\)), the \(t\)s cancel out and you get \(c^2\). This is an easy way to check if you've remembered the formulas correctly!
Key Takeaway
For a rectangular hyperbola \(xy = c^2\), the general point is always \((ct, \frac{c}{t})\). Unlike the parabola, this graph has two separate parts (branches) and never touches the \(x\) or \(y\) axes.
4. Tangents and Normals
A tangent is a straight line that just touches the curve at one point. A normal is a line perpendicular (at 90 degrees) to the tangent at that same point.
Finding the Gradient
To find the equation of these lines, we need the gradient (\(\frac{dy}{dx}\)).
For the Parabola (\(y^2 = 4ax\)):
Using differentiation, we find that \(\frac{dy}{dx} = \frac{2a}{y}\).
If you are using the parametric point \((at^2, 2at)\), the gradient simplifies beautifully to \(\frac{1}{t}\).
For the Rectangular Hyperbola (\(xy = c^2\)):
The gradient \(\frac{dy}{dx}\) at the point \((ct, \frac{c}{t})\) is \(-\frac{1}{t^2}\).
Condition for \(y = mx + c\) to be a Tangent
Sometimes the exam will ask if a specific line \(y = mx + c\) is a tangent to a parabola. For the parabola \(y^2 = 4ax\), the line is a tangent if \(c = \frac{a}{m}\).
Step-by-Step: Finding a Tangent Equation
1. Identify your point (either coordinates or in terms of \(t\)).
2. Find the gradient (\(m\)) at that point using the formulas above.
3. Use the straight-line formula: \(y - y_1 = m(x - x_1)\).
4. Simplify to the required form.
5. Loci Problems
A Locus (plural: Loci) is simply a set of points that satisfy a specific rule. In this chapter, you might be asked to find the path (the equation) traced out by the midpoint of a moving line or the intersection of two tangents.
How to approach Loci problems:
- Step 1: Write down the coordinates of the point you are interested in (e.g., the midpoint) using the parameters of the points you know.
- Step 2: You will now have equations for the "New \(X\)" and "New \(Y\)" in terms of \(t\).
- Step 3: Eliminate \(t\)! Rearrange one equation to make \(t\) the subject, then sub it into the other.
- Step 4: The resulting equation in \(x\) and \(y\) is your Locus. It will often turn out to be another parabola or hyperbola!
Don't worry if this seems tricky at first! The "Eliminate \(t\)" step is the most important part. Once \(t\) is gone, the algebra usually falls into place.
Summary: Your "Cheat Sheet" for Coordinate Systems
Parabola:
- Equation: \(y^2 = 4ax\)
- Point: \((at^2, 2at)\)
- Focus: \((a, 0)\), Directrix: \(x = -a\)
- Tangent Gradient: \(\frac{1}{t}\)
Rectangular Hyperbola:
- Equation: \(xy = c^2\)
- Point: \((ct, \frac{c}{t})\)
- Tangent Gradient: \(-\frac{1}{t^2}\)
Final Tip: Always keep a sketch in mind. Parabolas look like bowls; hyperbolas look like two mirroring boomerangs. Visualizing the shape helps you check if your answers make sense!