Welcome to Coordinate Geometry (FP1.2)!
In this chapter, we are going to become mathematical detectives. Instead of just looking at a finished shape, we are going to find the "rule" or "path" that a moving point follows. This path is called a locus.
Why does this matter? Understanding how points move in relation to other points and lines is the secret behind how satellite dishes focus signals, how car headlights reflect light, and even how planets orbit the sun! Don't worry if it seems abstract at first—we will break it down step-by-step.
1. What exactly is a "Locus"?
A locus (plural: loci) is simply a set of points that all follow the same rule.
Analogy: Imagine a dog tied to a pole with a 3-metre leash. If the dog walks around the pole keeping the leash tight, the path it draws is a circle. The "rule" is: "Stay exactly 3 metres from the pole." The circle is the locus.
Quick Review: The Tools You Need
Before we start, make sure you remember the Distance Formula. To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use:
\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
2. Distance from a Point vs. Distance from a Line
In this section of the syllabus, we focus on rules where a point must stay equidistant (the same distance) from a specific point and a specific straight line.
Distance from a Point
If we have a fixed point \(F(h, k)\) and a moving point \(P(x, y)\), the distance between them is always:
\(\sqrt{(x - h)^2 + (y - k)^2}\)
Distance from a Line
We only need to worry about vertical or horizontal lines here:
- The distance from point \((x, y)\) to a vertical line \(x = a\) is simply the horizontal difference: \(|x - a|\).
- The distance from point \((x, y)\) to a horizontal line \(y = b\) is simply the vertical difference: \(|y - b|\).
Key Takeaway: When finding a locus, we usually set these two distances equal to each other and solve the equation!
3. Step-by-Step: Finding the Cartesian Equation
Let’s look at the example mentioned in your syllabus: Find the Cartesian equation of the locus of points that are equidistant from the point \((2, 3)\) and the line \(x = 4\).
Step 1: Define your moving point
Let the moving point be \(P(x, y)\). This represents any point on our mystery path.
Step 2: Write the distance to the fixed point
The distance from \(P(x, y)\) to the point \((2, 3)\) is:
\(\sqrt{(x - 2)^2 + (y - 3)^2}\)
Step 3: Write the distance to the line
The distance from \(P(x, y)\) to the line \(x = 4\) is:
\(|x - 4|\)
Step 4: Set them equal and square both sides
Since the point is "equidistant," we write:
\(\sqrt{(x - 2)^2 + (y - 3)^2} = |x - 4|\)
To get rid of the square root, we square both sides:
\((x - 2)^2 + (y - 3)^2 = (x - 4)^2\)
Step 5: Expand and simplify
Expand the \(x\) terms:
\(x^2 - 4x + 4 + (y - 3)^2 = x^2 - 8x + 16\)
Notice how the \(x^2\) terms cancel out! Let's move everything to one side to find the equation for \(x\):
\(-4x + 4 + (y - 3)^2 = -8x + 16\)
\(4x = 12 - (y - 3)^2\)
\(x = 3 - \frac{1}{4}(y - 3)^2\)
Did you know?
The shape you just created is a parabola! In this context, the point \((2, 3)\) is called the focus and the line \(x = 4\) is called the directrix.
4. Common Pitfalls to Avoid
Don't worry if the algebra feels heavy at first. Here are the most common "trip-up" points:
- Forgetting to square the whole side: When you square \(|x-4|\), it becomes \((x-4)^2\). Don't just square the \(x\) and the \(4\) separately! Use FOIL/brackets: \((x-4)(x-4) = x^2 - 8x + 16\).
- Mixing up \(x\) and \(y\): If the line is \(x = a\), it's a vertical line. The distance depends only on the \(x\)-coordinate.
- Sign errors: Be very careful with minus signs when expanding \((x - (-2))^2\), which becomes \((x + 2)^2\).
5. Summary Quick Review
The "Locus Recipe":
- Start with \(P(x, y)\).
- Use the distance formula for the point: \(\sqrt{(x-h)^2 + (y-k)^2}\).
- Find the distance to the line: \(|x-a|\) or \(|y-b|\).
- Square both sides to remove the root.
- Simplify the algebra to get your final equation.
Key Takeaway: If the distance to a point equals the distance to a line, you are always finding the equation of a parabola. If the line is vertical (\(x=a\)), the parabola will open sideways. If the line is horizontal (\(y=b\)), it will open upwards or downwards.