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O-Level A-Math and SEC G3 Additional Mathematics K341

G2: Coordinate geometry in two dimensions

Connect lines, circles, distances, gradients, tangency, and loci through algebra and a labelled sketch.

Download this chapter PDF Download complete Additional Mathematics PDF

Core notes

Coordinate geometry converts geometric conditions into equations. Gradients establish parallel or perpendicular lines, coordinates give midpoints and areas, circle equations reveal centres and radii, and logarithmic transformations can turn selected nonlinear models into straight lines.

Lines, midpoints, and rectilinear area

Non-vertical parallel lines have equal gradients. Non-vertical perpendicular lines have gradient product $-1$ . The midpoint averages corresponding coordinates. For a rectilinear figure, divide it into triangles or trapezia, or use a coordinate-area method while keeping vertices in boundary order.

Coordinate geometry of circles

The standard form $(x-a)^2+(y-b)^2=r^2$ has centre $(a,b)$ and radius $r$ . Complete the square in $x$ and $y$ to convert $x^2+y^2+2gx+2fy+c=0$ , giving centre $(-g,-f)$ and $r^2=g^2+f^2-c$ . Problems involving two circles are excluded from this syllabus scope.

Linearising nonlinear relationships

For $y=ax^n$ , logarithms give $\log y=\log a+n\log x$ , so a graph of $\log y$ against $\log x$ has gradient $n$ and intercept $\log a$ . For $y=kb^x$ , $\log y=\log k+x\log b$ , so plotting $\log y$ against $x$ gives gradient $\log b$ and intercept $\log k$ . Use one logarithm base consistently.

Formulae and methods

Relationship	Use
$m_1=m_2$	Test non-vertical parallel lines.
$m_1m_2=-1$	Test non-vertical perpendicular lines.
$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$	Find the midpoint of a segment.
$(x-a)^2+(y-b)^2=r^2$	Read or construct a circle with centre $(a,b)$ .

Worked examples

Example 1: Find the centre and radius of $x^2+y^2-6x+4y-12=0$ .

Group and complete squares: $(x-3)^2-9+(y+2)^2-4-12=0$ .
Rearrange to $(x-3)^2+(y+2)^2=25$ .
Read the centre and positive radius.

Answer: Centre $(3,-2)$ , radius $5$ .

Example 2: A relationship $y=ax^n$ gives a straight line of gradient $2.5$ and vertical intercept $0.7$ on a $\log_{10}y$ against $\log_{10}x$ graph. Find $a$ and $n$ .

Compare $\log_{10}y=\log_{10}a+n\log_{10}x$ .
The gradient gives $n=2.5$ .
The intercept gives $\log_{10}a=0.7$ , so $a=10^{0.7}$ .

Answer: $a=10^{0.7}$ and $n=2.5$ .

Chapter checkpoint

Draw and label a sketch before choosing a gradient, distance, area, or circle relation.
Complete the square carefully when converting a general circle equation.
State the transformed axes, gradient, and intercept before extracting model constants.

Exam traps and retrieval check

Avoid these traps

Using gradient product $-1$ when one of the lines is vertical.
Reading the circle centre with the same signs as the completed-square brackets.
Extracting $a$ or $b$ directly from a logarithmic intercept or gradient without reversing the logarithm.

Check from memory

What is the centre of

(x+4)^2+(y-1)^2=9

$(-4,1)$ .

What does the gradient represent on a

\log y

against

\log x

graph for

y=ax^n

$n$ .

How is a midpoint found?

Average the two x-coordinates and average the two y-coordinates.

Official K341 coverage

This chapter maps to 5 official outcomes: G2.1, G2.2, G2.3, G2.4, G2.5. Check the official 2027 K341 syllabus for exact assessable wording and paper details.

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O-Level A-Math and SEC G3 Additional Mathematics K341

G2: Coordinate geometry in two dimensions

Connect lines, circles, distances, gradients, tangency, and loci through algebra and a labelled sketch.

Download this chapter PDF Download complete Additional Mathematics PDF

Core notes

Lines, midpoints, and rectilinear area

Coordinate geometry of circles

Linearising nonlinear relationships

Formulae and methods

Relationship	Use
$m_1=m_2$	Test non-vertical parallel lines.
$m_1m_2=-1$	Test non-vertical perpendicular lines.
$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$	Find the midpoint of a segment.
$(x-a)^2+(y-b)^2=r^2$	Read or construct a circle with centre $(a,b)$ .

Worked examples

Example 1: Find the centre and radius of $x^2+y^2-6x+4y-12=0$ .

Group and complete squares: $(x-3)^2-9+(y+2)^2-4-12=0$ .
Rearrange to $(x-3)^2+(y+2)^2=25$ .
Read the centre and positive radius.

Answer: Centre $(3,-2)$ , radius $5$ .

Example 2: A relationship $y=ax^n$ gives a straight line of gradient $2.5$ and vertical intercept $0.7$ on a $\log_{10}y$ against $\log_{10}x$ graph. Find $a$ and $n$ .

Compare $\log_{10}y=\log_{10}a+n\log_{10}x$ .
The gradient gives $n=2.5$ .
The intercept gives $\log_{10}a=0.7$ , so $a=10^{0.7}$ .

Answer: $a=10^{0.7}$ and $n=2.5$ .

Chapter checkpoint

Draw and label a sketch before choosing a gradient, distance, area, or circle relation.
Complete the square carefully when converting a general circle equation.
State the transformed axes, gradient, and intercept before extracting model constants.

Exam traps and retrieval check

Avoid these traps

Using gradient product $-1$ when one of the lines is vertical.
Reading the circle centre with the same signs as the completed-square brackets.
Extracting $a$ or $b$ directly from a logarithmic intercept or gradient without reversing the logarithm.

Check from memory

What is the centre of

(x+4)^2+(y-1)^2=9

$(-4,1)$ .

What does the gradient represent on a

\log y

against

\log x

graph for

y=ax^n

$n$ .

How is a midpoint found?

Average the two x-coordinates and average the two y-coordinates.

Official K341 coverage

This chapter maps to 5 official outcomes: G2.1, G2.2, G2.3, G2.4, G2.5. Check the official 2027 K341 syllabus for exact assessable wording and paper details.

Core notes

Lines, midpoints, and rectilinear area

Coordinate geometry of circles

Linearising nonlinear relationships

Formulae and methods

Worked examples

Example 1: Find the centre and radius of x2+y2−6x+4y−12=0x^2+y^2-6x+4y-12=0x2+y2−6x+4y−12=0.

Example 2: A relationship y=axny=ax^ny=axn gives a straight line of gradient 2.52.52.5 and vertical intercept 0.70.70.7 on a log⁡10y\log_{10}ylog10​y against log⁡10x\log_{10}xlog10​x graph. Find aaa and nnn.

Chapter checkpoint

Exam traps and retrieval check

Avoid these traps

Check from memory

Official K341 coverage

Core notes

Lines, midpoints, and rectilinear area

Coordinate geometry of circles

Linearising nonlinear relationships

Formulae and methods

Worked examples

Example 1: Find the centre and radius of x2+y2−6x+4y−12=0x^2+y^2-6x+4y-12=0x2+y2−6x+4y−12=0.

Example 2: A relationship y=axny=ax^ny=axn gives a straight line of gradient 2.52.52.5 and vertical intercept 0.70.70.7 on a log⁡10y\log_{10}ylog10​y against log⁡10x\log_{10}xlog10​x graph. Find aaa and nnn.

Chapter checkpoint

Exam traps and retrieval check

Avoid these traps

Check from memory

Official K341 coverage

Example 1: Find the centre and radius of $x^2+y^2-6x+4y-12=0$ .

Example 2: A relationship $y=ax^n$ gives a straight line of gradient $2.5$ and vertical intercept $0.7$ on a $\log_{10}y$ against $\log_{10}x$ graph. Find $a$ and $n$ .

Example 1: Find the centre and radius of $x^2+y^2-6x+4y-12=0$ .

Example 2: A relationship $y=ax^n$ gives a straight line of gradient $2.5$ and vertical intercept $0.7$ on a $\log_{10}y$ against $\log_{10}x$ graph. Find $a$ and $n$ .