Coordinate systems - Further Mathematics - Pearson A-Level

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Welcome to Coordinate Systems in FP1!

Hello there! Coordinate geometry might feel familiar from previous studies, but in Further Mathematics 1 (FP1), we sharpen those tools and apply them to more challenging concepts, especially when dealing with conic sections (like parabolas and hyperbolas).

This chapter is about mastering the algebraic rules governing points, lines, and areas on the 2D plane. Think of these skills as your fundamental toolkit for all the geometric problems ahead. Don't worry if some concepts seem tricky at first; we will break them down step-by-step!

Why is this important?

It provides the framework for understanding loci (the path of a moving point).
It is essential for finding properties of conics later in the course.
It allows us to translate visual geometry into precise algebraic equations.

Section 1: The Essential Toolkit Refresher

While these concepts are prerequisites, a quick refresher ensures our foundation is solid.

1.1 Distance Between Two Points

If you have two points, $A(x_1, y_1)$ and $B(x_2, y_2)$, the distance $d$ between them is found using Pythagoras’ Theorem:

$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

1.2 Midpoint of a Line Segment

The midpoint $M$ is simply the average of the coordinates:

$$ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$

1.3 Gradient (Slope) of a Line

The gradient $m$ measures the steepness of the line connecting A and B:

$$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{Change in } y}{\text{Change in } x} $$

Quick Review Box: These three formulas are the absolute minimum you must have memorized before tackling FP1 problems.

Section 2: Working with Lines and Intersections

In FP1, we often need to find where lines meet, especially when determining the intersection of a line with a conic section.

2.1 Equation of a Straight Line

We mainly use the point-gradient form, which is highly effective:

$$ y - y_1 = m(x - x_1) $$

where $m$ is the gradient and $(x_1, y_1)$ is any point on the line.

2.2 Finding the Point of Intersection

To find where two lines, $L_1$ and $L_2$, intersect, you must solve their equations simultaneously.

Step-by-step Process:

Ensure both line equations are simplified (often into the form $Ax + By = C$).
Use the Substitution Method or the Elimination Method to solve for $x$ and $y$.
Once you have $x$, substitute it back into one of the original equations to find $y$.

Did You Know? If the simultaneous equations yield a contradiction (e.g., $0 = 5$), the lines are parallel and never intersect. If they yield an identity (e.g., $0 = 0$), the lines are the same (coincident).

2.3 Perpendicular Lines: The Golden Rule

Two lines are perpendicular (or orthogonal) if the product of their gradients is $-1$.

If $m_1$ is the gradient of $L_1$ and $m_2$ is the gradient of $L_2$, then:

$$ m_1 m_2 = -1 \quad \text{or} \quad m_2 = - \frac{1}{m_1} $$

Memory Aid: The gradient of a perpendicular line is the negative reciprocal of the original gradient. Flip the fraction and change the sign!

Example: If a line has a gradient of $m_1 = \frac{2}{3}$, the perpendicular line has a gradient of $m_2 = -\frac{3}{2}$.

Common Mistake Alert! Students often forget the negative sign. A gradient of 3 has a perpendicular gradient of $-1/3$, not just $1/3$.

Section 3: Calculating the Area of a Triangle

In FP1, simply using $ \frac{1}{2} \times \text{base} \times \text{height} $ can be very complicated if the base and height are not aligned with the axes. We use a powerful coordinate method, often derived from matrix determinants, to find the area directly from the vertices.

The Coordinate Area Formula (Shoelace Method)

Given the vertices of a triangle $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$, the area $K$ is given by:

$$ K = \frac{1}{2} \left| (x_1 y_2 + x_2 y_3 + x_3 y_1) - (y_1 x_2 + y_2 x_3 + y_3 x_1) \right| $$

Don't worry, that formula looks complicated, but there’s a simple visual way to set it up!

Step-by-step Process for Area Calculation

Step 1: List the Coordinates
Write the coordinates in two columns, repeating the first coordinate pair at the end:

$$ \begin{matrix} x_1 & y_1 \\ x_2 & y_2 \\ x_3 & y_3 \\ x_1 & y_1 \end{matrix} $$

Step 2: Multiply Diagonally (Downward/Right)
Multiply the terms diagonally down (as shown by the red arrows in the analogy below) and sum them up. Let this sum be $S_1$.

$$ S_1 = x_1 y_2 + x_2 y_3 + x_3 y_1 $$

Step 3: Multiply Diagonally (Upward/Right)
Multiply the terms diagonally up (as shown by the blue arrows in the analogy below) and sum them up. Let this sum be $S_2$.

$$ S_2 = y_1 x_2 + y_2 x_3 + y_3 x_1 $$

Analogy: The Shoelace Method

Imagine connecting the coordinates like tying a shoelace.
Downwards (Red): $x_1 \to y_2$, $x_2 \to y_3$, $x_3 \to y_1$
Upwards (Blue): $y_1 \to x_2$, $y_2 \to x_3$, $y_3 \to x_1$

Step 4: Calculate the Area
The area $K$ is half the absolute difference between the two sums:

$$ K = \frac{1}{2} | S_1 - S_2 | $$

Important Note: The vertical bars $ | \ldots | $ mean absolute value. Area must always be positive. If $S_1 - S_2$ results in $-10$, the area is $5$.

Key Takeaway for Section 3

This method is powerful and applies not just to triangles, but to any polygon. Ensure you list the vertices in order (clockwise or anti-clockwise) and remember to repeat the first vertex at the end.

Section 4: Loci and Parametric Coordinates

The most significant use of coordinate geometry in FP1 is describing loci (plural of locus) and using parametric equations.

4.1 What is a Locus?

A locus is the set of all points that satisfy a specific geometric condition.

Example: The locus of a point 5 units away from the origin is a circle with radius 5.

When solving locus problems, you usually define a general point $P(x, y)$ and use the given condition (e.g., distance, ratio of distances) to create an equation linking $x$ and $y$. This resultant equation is the Cartesian equation of the locus.

4.2 Parametric Equations

When dealing with curves like parabolas or hyperbolas, defining points using standard $y = f(x)$ form can be cumbersome. Instead, we use a third variable, called the parameter (often $t$), to define both $x$ and $y$.

$$ x = f(t) \quad \text{and} \quad y = g(t) $$

For example, a point P on a rectangular hyperbola might be defined by:

$$ x = 3t, \quad y = \frac{3}{t} $$

Why use Parametric Equations?
They simplify geometric calculations (like finding the gradient of a tangent) and allow us to describe complicated curves much more elegantly than Cartesian coordinates alone.

4.3 Converting Parametric to Cartesian Form (Eliminating the Parameter)

A core skill in this chapter is converting between the two forms. This involves using substitution to eliminate the parameter $t$.

Step-by-step Example: Convert $x = 3t$ and $y = \frac{3}{t}$ to Cartesian form.

Isolate $t$ in the simpler equation: $$ t = \frac{x}{3} $$
Substitute this expression for $t$ into the second equation: $$ y = \frac{3}{\left(\frac{x}{3}\right)} $$
Simplify the expression: $$ y = 3 \times \frac{3}{x} $$ $$ y = \frac{9}{x} \quad \text{or} \quad xy = 9 $$

This is the Cartesian equation for the hyperbola described parametrically.

Encouraging Tip: If the parameter involves trigonometric functions (like $t = \sin \theta$), you often eliminate $t$ using trigonometric identities, such as $\sin^2 \theta + \cos^2 \theta = 1$.

Key Takeaway for Section 4

Parametric coordinates are your friend in FP1. Practice eliminating the parameter—this skill is vital when defining the Cartesian equation of the conics you study later.

Section 5: Study Tips and Common Pitfalls

Avoid These Common Mistakes

Forgetting the Absolute Value: When calculating area using the coordinate method, you must take $ \frac{1}{2} | S_1 - S_2 | $. Area cannot be negative.
Misunderstanding Perpendicularity: Always use the negative reciprocal. A gradient of $-5$ becomes $\frac{1}{5}$.
Algebra Errors in Elimination: When solving simultaneous equations or eliminating the parameter $t$, be extremely careful with sign changes and fractions.
Repeating the First Point: Ensure you include the first point at the end of your list when using the Shoelace/Coordinate Area method (Step 1 in Section 3).

Tips for Success

Sketch Everything: Whenever possible, draw a quick sketch of the points and lines given. This helps you spot obvious errors (e.g., if you calculate a positive gradient but your sketch shows a line sloping downwards).
Systematic Approach: Write down all known coordinates and formulas before you start calculating. This reduces calculation errors under exam pressure.
Practice Parameter Elimination: Spend dedicated time practicing different types of elimination (algebraic, fractional, trigonometric).

You have successfully reviewed the essential tools of coordinate geometry required for FP1. Keep practicing these foundational skills, and you will be well-prepared to tackle the challenging conics material ahead! Keep up the great work!