Coordinate Geometry

Learn, then practise

The original worked lessons remain the main explanations. Use these focused reminders to check the conditions before you start.

Distance, midpoint and geometric reasoning

Calculate exact distances and full midpoint coordinates.
Recover missing endpoints and all permitted points.
Use squared lengths and midpoint properties to justify geometric claims.

For \(A(x_1,y_1)\), \(B(x_2,y_2)\): \(AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\), \(\displaystyle M= \bigl (\frac{x_{1}+ x_{2}}{2}, \frac{y_{1}+ y_{2}}{2}\bigr )\). Equal distances alone do not make a point a midpoint: it must also lie on the segment.

Read the full notes and 3 worked examples · Next: short practice

Parallel, perpendicular and exceptional lines

Use equal gradients and negative reciprocals correctly.
Handle horizontal and vertical lines without division by zero.
Distinguish coincident lines from distinct parallel lines; construct perpendicular bisectors.

For nonvertical lines, parallel directions have equal gradients. For two nonvertical perpendicular lines, \(m_1m_2=-1\). A vertical line \(x=a\) is perpendicular to a horizontal line \(y=b\). Equal gradients do not establish that two lines are distinct.

Read the full notes and 3 worked examples · Next: short practice

Straight-line equations, intercepts and intersections

Build a complete equation from sufficient information.
Find paired intersection coordinates and interpret intercepts.
Handle parameter cases with zero, one or infinitely many intersections.

Use \(y-y_0=m(x-x_0)\), then check the known point. For an x-intercept set \(y=0\); for a y-intercept set \(x=0\). Both original equations must hold at an intersection. An intercept triangle has area \(\tfrac12|ab|\), where \(a,b\) are the nonzero axis intercepts.

Read the full notes and 3 worked examples · Next: short practice

Circle equations and tangents

Connect completed-square and expanded circle equations.
Find circles from a centre, a diameter or sufficient points.
Use the radius–tangent right angle, including exceptional directions.

\((x-a)^2+(y-b)^2=r^2\) has centre \((a,b)\) and radius \(r>0\). Complete both squares; divide by any common coefficient first. A tangent at a point on the circle is perpendicular to the radius. \(r^2=0\) is one point, not a nondegenerate circle.

Read the full notes and 3 worked examples · Next: short practice

Line–circle intersections, chords and tangency

Solve for all paired line–circle intersections and filter segment or ray restrictions.
Use a repeated root or radius geometry to justify tangency.
Interpret discriminants, chord lengths and line–quadratic-curve intersections.

Substitute the line into the curve. A quadratic with positive, zero or negative discriminant gives two distinct, one repeated or no real intersection coordinates. Recover the matching other coordinate. A segment or ray can contain fewer points than its supporting line. A perpendicular from the centre bisects a chord.

Read the full notes and 3 worked examples · Next: short practice