Coordinate Geometry

Coordinate geometry translates geometric problems into algebra that can be solved with equations. At this level it covers two main areas: straight lines (extending CSEC Maths) and circles (new at Additional Mathematics level).

Key Formulas¶

For two points $A(x_1, y_1)$ and $B(x_2, y_2)$:

The distance formula is Pythagoras applied to the horizontal and vertical separations.

Equations of Straight Lines¶

A straight line can be written in several equivalent forms:

• Slope-intercept: $y = mx + c$ (gradient $m$, $y$-intercept $c$)
• Point-slope: $y - y_1 = m(x - x_1)$ (given gradient $m$ and one point)
• General: $ax + by + c = 0$

Finding the equation given two points: calculate the gradient first, then substitute one point into point-slope form.

Parallel and Perpendicular Lines¶

• Parallel lines have equal gradients: $m_1 = m_2$.
• Perpendicular lines have gradients whose product is $-1$: $m_1 m_2 = -1$, equivalently $m_2 = -1/m_1$.

Perpendicular Bisector¶

The perpendicular bisector of a line segment $AB$ passes through the midpoint of $AB$ and is perpendicular to $AB$.

Intersection of Two Straight Lines¶

To find the point where two lines meet, solve their equations simultaneously — set the expressions for $y$ equal and solve for $x$, then back-substitute.

Circles¶

Standard Form¶

The equation of a circle with centre $(a, b)$ and radius $r$:

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

Every point on the circle is exactly $r$ units from the centre, so this equation comes directly from the distance formula.

General Form¶

Expanding the standard form gives:

$x^2 + y^2 + 2fx + 2gy + c = 0$

In this form the centre is $(-f, -g)$ and the radius is $\sqrt{f^2 + g^2 - c}$, but you do not need to memorise these formulas. It is safer and more reliable to complete the square for both $x$ and $y$, which converts general form back to standard form directly.

Tangents and Normals to Circles¶

A tangent to a circle at a point $P$ is perpendicular to the radius at $P$.

Method for finding the tangent at $P$:

1. Find the gradient of the radius from the centre to $P$.
2. The tangent gradient is the negative reciprocal.
3. Write the tangent equation using point-slope form with point $P$.

The normal at $P$ passes through the centre. Its equation uses the radius gradient (not the perpendicular gradient).

Intersection of a Line and a Circle¶

Substitute the line's equation into the circle's equation to get a quadratic in one variable. The discriminant of that quadratic tells you about the intersection:

• $\Delta > 0$: two intersection points (line is a secant).
• $\Delta = 0$: one intersection point (line is a tangent).
• $\Delta < 0$: no intersection.

Collinearity¶

Three points $A$, $B$, $C$ are collinear if the gradient of $AB$ equals the gradient of $BC$ (or equivalently, the gradient of $AC$). Alternatively, show that $\overrightarrow{AB}$ is a scalar multiple of $\overrightarrow{AC}$.