Mathematics (9709) | Coordinate geometry

🧠 Pure Mathematics 1 (9709) Study Notes: Coordinate Geometry (1.3)

Hey there, future mathematician! Welcome to Coordinate Geometry. This chapter is super important because it takes the algebra you already know and connects it directly to shapes and diagrams you can visualize.

Think of this chapter as learning the mathematical rules for reading a map. When you know these rules, you can calculate distances, find halfway points, and describe exactly where a straight line or a curve (like a circle) sits in space. Mastery here builds the foundation for topics like Differentiation and Integration!

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Section 1: The Essential Straight Line

The foundation of coordinate geometry relies on calculating three fundamental properties between two points, A \((x_1, y_1)\) and B \((x_2, y_2)\).

1.1 Key Formulas for Points and Segments

A. Distance (Length)

This tells you how far apart the two points are. We use the Pythagoras theorem to find it.

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

Analogy: If you walk 3 units East (change in x) and 4 units North (change in y), the straight-line distance (the hypotenuse) is 5 units.

B. Midpoint

The midpoint is simply the average position between the two points.

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

C. Gradient (Slope), \(m\)

The gradient measures the steepness or slope of the line. It is the ratio of vertical change (rise) to horizontal change (run).

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Key Takeaway: If \(m\) is positive, the line slopes up (from left to right). If \(m\) is negative, the line slopes down.

1.2 Forms of the Straight Line Equation

You must be comfortable using three different forms of the line equation:

1. Gradient-Intercept Form: \(y = mx + c\)

• \(m\) is the gradient.
• \(c\) is the y-intercept (where the line crosses the y-axis).
• Best for sketching or quickly identifying the slope and intercept.

2. Point-Gradient Form: \(y - y_1 = m(x - x_1)\)

• This is the most efficient form for finding the equation of a line if you have the gradient \(m\) and any point \((x_1, y_1)\) on the line.

3. General Form: \(ax + by + c = 0\)

• \(a, b,\) and \(c\) are integers.
• Best for solving simultaneous equations (finding intersections).

1.3 Parallel and Perpendicular Lines

The gradient \(m\) is the key to understanding how lines relate geometrically.

A. Parallel Lines

Two lines are parallel if and only if they have the same gradient.

$$m_1 = m_2$$

B. Perpendicular Lines

Two lines are perpendicular (meet at a 90° angle) if the product of their gradients is \(-1\).

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

Memory Aid: To find the perpendicular gradient, you must "Flip it and negate it." (Find the reciprocal and change the sign).
Example: If \(m_1 = \frac{2}{3}\), then \(m_2 = -\frac{3}{2}\).

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Section 2: Lines and Curves Working Together

When solving problems involving a straight line and a curve (especially a quadratic or another line), you are usually asked to find where they cross, or to determine the conditions under which they cross.

2.1 Finding Points of Intersection

The points where two graphs intersect represent the solutions to their equations when solved simultaneously.

Step-by-step process:

1. Set the two equations equal to each other (usually substitute the linear equation, \(y = mx + c\), into the quadratic equation).
2. Rearrange the resulting equation into the standard quadratic form: \(Ax^2 + Bx + C = 0\).
3. Solve the quadratic equation for \(x\) (by factoring, formula, or completing the square).
4. Substitute the \(x\) value(s) back into the linear equation to find the corresponding \(y\) value(s).

Did you know? The solutions \((x, y)\) are the only points that satisfy both equations simultaneously.

2.2 Using the Discriminant for Intersections

If you have a quadratic equation \(Ax^2 + Bx + C = 0\), the discriminant, \(\Delta\), tells you about the nature of the roots (how many times the graphs intersect).

$$\Delta = B^2 - 4AC$$

• If \(\Delta > 0\): There are two distinct real roots. The line intersects the curve at two separate points.
• If \(\Delta = 0\): There is one repeated real root. The line is a tangent to the curve (it touches the curve at exactly one point).
• If \(\Delta < 0\): There are no real roots. The line does not intersect the curve at all.

This method is commonly used to find the range of values for an unknown constant (like \(k\)) in a line equation \(y = x + k\) such that it, for example, touches a quadratic curve.

🚫 Common Mistake Alert

When solving for intersections, always substitute the linear equation into the curve equation FIRST, then simplify into the \(Ax^2 + Bx + C = 0\) form before applying the discriminant. Do not use the coefficients from the original curve equation!

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Section 3: Exploring Circles

A circle is defined by its fixed centre and its fixed radius. You need to be able to use two forms of the circle equation and apply key geometric properties.

3.1 The Standard Circle Equation

If a circle has a centre \((a, b)\) and a radius \(r\), its equation is:

$$(x - a)^2 + (y - b)^2 = r^2$$

Example: A circle with centre \((3, -2)\) and radius 5 has the equation \((x - 3)^2 + (y + 2)^2 = 25\).

3.2 Finding the Centre and Radius (Completing the Square)

Sometimes the circle equation is given in the expanded form: \(x^2 + y^2 + 2gx + 2fy + c = 0\). To find the centre and radius, you must use Completing the Square.

Step-by-step process:

1. Group the \(x\) terms and \(y\) terms together:
   \((x^2 + 2gx) + (y^2 + 2fy) + c = 0\)
2. Complete the square for the \(x\) terms and \(y\) terms separately:
   \((x + g)^2 - g^2 + (y + f)^2 - f^2 + c = 0\)
3. Rearrange into the standard form \((x - a)^2 + (y - b)^2 = r^2\):
   $$(x + g)^2 + (y + f)^2 = g^2 + f^2 - c$$

From the final standard form, you can immediately identify:

• Centre: \((-g, -f)\)
• Radius: \(r = \sqrt{g^2 + f^2 - c}\)

Don't worry if this seems tricky at first. Practice completing the square for general equations, and it will become second nature!

3.3 Geometric Properties involving Circles and Lines

When solving problems involving tangents, intersections, or chords, rely on these foundational geometric facts:

• Tangent Perpendicular to Radius: A tangent line to a circle is always perpendicular to the radius drawn to the point of contact. (You use the \(m_1 m_2 = -1\) rule here!).
• Angle in a Semicircle: The angle subtended at the circumference by a diameter is always a right angle (\(90^\circ\)).
• Perpendicular Bisector of a Chord: The perpendicular bisector of any chord of a circle passes through the centre of the circle.

Example Use: If a question asks for the equation of the tangent at point P, you first find the gradient of the radius (Centre to P). Then, the tangent's gradient is the negative reciprocal of the radius gradient.