Hello Additional Mathematicians! Navigating the Coordinate Circle
Welcome to one of the most practical and exciting topics in Additional Mathematics: Coordinate Geometry of the Circle (Syllabus Topic 8). This chapter brings together everything you know about straight lines, gradients, and distances, and applies them to the perfect shape – the circle.
Don’t worry if coordinate geometry sometimes feels abstract. A circle is just a collection of points that are all the same distance from a fixed central point. Once we express this idea using algebra, all the complex problems become solvable!
The key to success here is mastering the two main forms of the circle equation and understanding the powerful relationship between a radius and a tangent.
1. Equations of a Circle: The Two Essential Forms
1.1 The Standard Form (Centre-Radius Form)
This is the most intuitive way to define a circle. If a circle has a centre at the point \((a, b)\) and a radius of \(r\), its equation is:
Formula (Given in the List of Formulas):
$$ (x – a)^2 + (y – b)^2 = r^2 $$
Analogy: Think of the centre \((a, b)\) as the origin point in a video game, and \(r\) as the radius of the "blast zone." Any point \((x, y)\) hit by the blast satisfies this equation.
How to use the Standard Form:
- If the centre is \((3, -1)\) and the radius is \(5\).
The equation is: \((x - 3)^2 + (y - (-1))^2 = 5^2\)
$$ (x - 3)^2 + (y + 1)^2 = 25 $$
Memory Aid: Notice the signs! If the equation has \((x - 3)\), the x-coordinate of the centre is positive 3. If it has \((y + 1)\), the y-coordinate of the centre is \(-1\).
1.2 The General Form
Sometimes, circle equations are given in an expanded, messy form. This is called the General Form:
$$ x^2 + y^2 + 2gx + 2fy + c = 0 $$Note: The coefficients of \(x^2\) and \(y^2\) must both be 1 in this form. If they are not 1 (e.g., \(2x^2 + 2y^2 + ...\)), you must divide the entire equation by that coefficient first!
Converting General Form to find Centre and Radius
To find the centre and radius from the General Form, you must use the method of Completing the Square.
However, you can also use these quick relationships derived from completing the square:
- Centre: \((-g, -f)\)
- Radius: $$ r = \sqrt{g^2 + f^2 - c} $$
Step-by-Step Example (Conversion):
Find the centre and radius of \(x^2 + y^2 - 6x + 4y - 12 = 0\).
- Identify \(2g = -6\) and \(2f = 4\), and \(c = -12\).
- Find \(g\) and \(f\): \(g = -3\), \(f = 2\).
- Centre \((-g, -f)\) is \((3, -2)\).
- Calculate Radius \(r\):
$$ r = \sqrt{(-3)^2 + (2)^2 - (-12)} $$ $$ r = \sqrt{9 + 4 + 12} = \sqrt{25} = 5 $$
Key Takeaway for Section 1: Master both circle equation forms and be lightning fast at finding the centre and radius, especially from the General Form, as this information is vital for solving intersection and tangent problems.
2. Intersections of a Straight Line and a Circle (Syllabus 8.2)
A straight line can interact with a circle in three possible ways: it can be a chord (intersecting twice), a tangent (intersecting once), or it might miss the circle entirely (no intersection).
2.1 Finding Points of Intersection
To find where a line (e.g., \(y = mx + k\)) meets a circle, you solve the equations simultaneously, usually by substitution.
Process:
- Substitute the linear equation (e.g., \(y\) isolated) into the circle equation.
- Expand and simplify the resulting equation. This will always result in a Quadratic Equation in one variable (usually \(x\)) in the form \(Ax^2 + Bx + C = 0\).
- Solve the quadratic equation for \(x\).
- Substitute the \(x\) values back into the linear equation to find the corresponding \(y\) values.
2.2 Using the Discriminant (\(b^2 - 4ac\))
If the question only asks how many times the line intersects the circle, you do not need to solve the quadratic completely. You only need to use the Discriminant (\(\Delta\)) from the resulting quadratic \(Ax^2 + Bx + C = 0\).
- Case 1: Chord
If \(\Delta > 0\) (\(b^2 - 4ac > 0\)), there are two distinct real roots. The line is a chord and intersects the circle at two points. - Case 2: Tangent
If \(\Delta = 0\) (\(b^2 - 4ac = 0\)), there is one repeated real root. The line is a tangent and touches the circle at exactly one point. - Case 3: No Intersection
If \(\Delta < 0\) (\(b^2 - 4ac < 0\)), there are no real roots. The line does not intersect the circle.
Did you know? This discriminant method is mathematically equivalent to calculating the perpendicular distance from the centre of the circle to the straight line and comparing it to the radius \(r\). However, using the discriminant after substitution is often faster in the exam!
Key Takeaway for Section 2: Intersection problems rely on standard simultaneous equation techniques that result in a quadratic. Use the discriminant to quickly determine the nature of the intersection.
3. Solving Problems Involving Tangents (Syllabus 8.3)
In Additional Mathematics, when finding the equation of a tangent to a circle, you must not use calculus (differentiation). We rely entirely on the essential geometric property of circles.
The Golden Rule for Tangents:
A radius drawn to the point of tangency is perpendicular to the tangent line at that point.
This means if the radius has a gradient \(m_{radius}\), the tangent must have a gradient \(m_{tangent}\) such that:
$$ m_{tangent} = -\frac{1}{m_{radius}} $$Step-by-Step: Finding the Equation of a Tangent
Suppose you are given the circle and a point \(P(x_1, y_1)\) on the circumference where the tangent passes.
- Find the Centre (C): Determine the coordinates \((a, b)\) of the circle's centre from its equation.
- Calculate the Gradient of the Radius (CP): Use the formula: $$ m_{radius} = \frac{y_1 - b}{x_1 - a} $$
- Calculate the Gradient of the Tangent: Use the perpendicular gradient rule: $$ m_{tangent} = -\frac{1}{m_{radius}} $$
- Find the Tangent Equation: Use the point-slope form:
$$ y - y_1 = m_{tangent} (x - x_1) $$
(Remember, \((x_1, y_1)\) is the given point on the circle.)
Common Mistake to Avoid:
If you are given the equation of a line and asked if it is a tangent, DO NOT assume it is perpendicular to the radius unless you first verify that the point of contact lies exactly on the circle. Always use the discriminant method (Section 2.2) to confirm tangency first!
Quick Review: Tangents
Requirement: Find centre \((a, b)\) and point of contact \((x_1, y_1)\).
1. Calculate \(m_{radius}\).
2. Calculate \(m_{tangent} = -1/m_{radius}\).
3. Use \(y - y_1 = m_{tangent}(x - x_1)\).
Key Takeaway for Section 3: The entire process of finding a tangent relies on geometry: the radius and tangent are perpendicular. If you get stuck, draw a diagram!
4. Intersections of Two Circles (Syllabus 8.4)
Just like a line and a circle, two circles can intersect (twice), touch (once), or not intersect at all.
4.1 Finding the Equation of the Common Chord
When two circles intersect, the straight line connecting the two intersection points is called the common chord.
Consider two circles with equations:
Circle 1: $$ x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0 $$ Circle 2: $$ x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0 $$
Any point \((x, y)\) that satisfies both equations must lie on the common chord.
Process: To find the equation of the common chord, simply subtract the two circle equations from each other.
$$ (x^2 + y^2 + 2g_1x + 2f_1y + c_1) - (x^2 + y^2 + 2g_2x + 2f_2y + c_2) = 0 $$
Since the \(x^2\) and \(y^2\) terms cancel out, the result is always a linear equation (a straight line) which is the equation of the common chord.
Example: If Circle 1 is \(x^2 + y^2 - 4x = 0\) and Circle 2 is \(x^2 + y^2 - 6y - 7 = 0\).
Subtracting (C1 - C2) gives: \((-4x) - (-6y - 7) = 0\).
Common Chord Equation: \(-4x + 6y + 7 = 0\).
4.2 Determining Intersections of Two Circles
To determine if two circles intersect, touch, or miss, we look at the distance between their centres (\(d\)) compared to the sum of their radii (\(r_1 + r_2\)).
Step-by-Step Analysis:
- Find the centre \((a_1, b_1)\) and radius \(r_1\) for Circle 1.
- Find the centre \((a_2, b_2)\) and radius \(r_2\) for Circle 2.
- Calculate the distance \(d\) between the two centres using the distance formula: $$ d = \sqrt{(a_2 - a_1)^2 + (b_2 - b_1)^2} $$
- Compare \(d\) with \(r_1 + r_2\):
- Intersect (Two points): If \(d < r_1 + r_2\). (The centres are close enough that the circles overlap.)
- Touch (One point, external tangency): If \(d = r_1 + r_2\). (The circles touch externally.)
- Do Not Intersect: If \(d > r_1 + r_2\). (The circles are too far apart.)
Note: There is also a case of internal tangency where \(d = |r_1 - r_2|\), and one circle is entirely inside the other, touching at one point. This is also categorized under "touch" or "one point of intersection."
Key Takeaway for Section 4: Finding the common chord is an easy subtraction trick. Determining the relationship relies on comparing the distance between the centres with the combined radius.
Chapter Summary: Must-Know Formulas
Circle Equation (Standard Form)
$$ (x – a)^2 + (y – b)^2 = r^2 $$Centre \((a, b)\), Radius \(r\).
Circle Equation (General Form)
$$ x^2 + y^2 + 2gx + 2fy + c = 0 $$Centre \((-g, -f)\), Radius \(r = \sqrt{g^2 + f^2 - c}\).
Line Intersections (Discriminant)
- Two points: \(b^2 - 4ac > 0\)
- Tangent: \(b^2 - 4ac = 0\)
- No intersection: \(b^2 - 4ac < 0\)
Tangents
Use the perpendicular gradient rule: \(m_{tangent} = -1/m_{radius}\).
You've got this! Coordinate geometry of the circle is very structured. If you learn these core formulas and geometric rules, you will master this chapter quickly!