Coordinate geometry

Explanation

Scope and a diagram-free coordinate toolkit

Coordinate geometry translates geometric facts into equations. A point is represented by an ordered pair, a line by a linear equation, and a circle by a quadratic equation in x and y. Intersections are coordinates that satisfy both relevant equations, so algebra can replace an accurately drawn diagram.

The main habits in this pack are to state the formula being used, preserve exact values such as square roots, and substitute final coordinates back into every original equation. When a question asks for a distance or radius, the result is non-negative; when it asks for a point, both coordinates are required.

This pack is limited to Cambridge 9709 Pure Mathematics 1 section 1.3. It includes elementary geometry of circles but excludes implicit differentiation, and it does not claim to cover other Paper 1 sections or a complete AS Mathematics route.

Explanation

Gradient and the three useful line forms

For distinct points (x1, y1) and (x2, y2) with x2 not equal to x1, the gradient is m = (y2 - y1)/(x2 - x1). The order of subtraction must match in numerator and denominator. If x2 = x1, the line is vertical and its gradient is undefined; its equation is x = x1.

The form y = mx + c displays gradient m and y-intercept c. The point-gradient form y - y1 = m(x - x1) is often fastest when one point and the gradient are known. The general form ax + by + c = 0 is useful for intersections and also includes vertical lines. If b is non-zero, rearranging shows its gradient is -a/b.

To find a line from two points, calculate the gradient, substitute either point into point-gradient form, and then rearrange only if another form is requested. Checking both given points in the final equation detects sign and intercept errors.

Explanation

Parallel, perpendicular, horizontal and vertical lines

Distinct non-vertical parallel lines have equal gradients. Non-vertical perpendicular lines have gradients whose product is -1, so a line of gradient m, where m is non-zero, has perpendicular gradient -1/m. Equal gradients alone do not prove that two equations name distinct lines; they may describe the same line.

The special cases matter. A horizontal line has equation y = constant and gradient 0. A vertical line has equation x = constant and undefined gradient. Horizontal and vertical lines are perpendicular even though multiplying their gradients is not possible.

A perpendicular bisector passes through a segment's midpoint and is perpendicular to the segment. Find the midpoint first, then use the negative reciprocal gradient unless the segment is horizontal or vertical, in which case use the special-case equations.

Explanation

Midpoints, distances and intersections

The midpoint of A(x1, y1) and B(x2, y2) is ((x1 + x2)/2, (y1 + y2)/2). The distance AB is sqrt((x2 - x1)^2 + (y2 - y1)^2), obtained from Pythagoras. Squaring removes sign differences, so reversing the order of the two points gives the same distance.

To find an intersection of two lines, solve their equations simultaneously. Substitution is convenient when one variable is already isolated; elimination is often efficient in general form. A single solution means one intersection, inconsistent parallel equations mean no intersection, and equivalent equations represent the same line with infinitely many common points.

Do not infer a midpoint, distance or intersection from appearance. All values should follow from the stated coordinates and equations, and an intersection should be checked in both original equations.

Explanation

Standard and expanded equations of circles

The equation (x - a)^2 + (y - b)^2 = r^2 represents the circle with centre (a, b) and radius r, where r is positive. Notice the reversed signs inside the brackets: (x + 3)^2 corresponds to centre x-coordinate -3.

An expanded circle has the form x^2 + y^2 + 2gx + 2fy + c = 0. Completing the square gives (x + g)^2 + (y + f)^2 = g^2 + f^2 - c, so the centre is (-g, -f) and the radius is sqrt(g^2 + f^2 - c), provided this quantity is positive.

Equal coefficients of x^2 and y^2 and no xy term are characteristic of these axis-aligned circle equations. Completing both squares and expanding back again is a reliable sign check.

Explanation

Points, symmetry, diameters and the angle in a semicircle

For a circle with centre C and radius r, a point P lies on the circle exactly when CP^2 = r^2. A smaller squared distance places P inside and a larger squared distance places it outside. Comparing squared distances avoids unnecessary square roots.

The centre is the midpoint of every diameter. If the endpoints of a diameter are known, their midpoint gives the centre and half their distance gives the radius. Conversely, reflection in the centre maps one end of a diameter to the other.

An angle subtended by a diameter at any other point on the circle is 90 degrees. This result may justify perpendicular sides in a triangle, but the point must be stated to lie on the circle and the chord used must actually be a diameter.

Explanation

Algebraic intersections of a line and a circle

To intersect a line and a circle, express one coordinate from the line and substitute it into the circle. The resulting quadratic can have two distinct real roots, one repeated root or no real roots, corresponding respectively to a secant, a tangent or a line that misses the circle.

Each root gives one coordinate; substitute back into the line to recover its paired coordinate. Do not combine an x-coordinate from one root with the y-coordinate belonging to another. Check each ordered pair in both original equations.

A discriminant argument can count intersections without finding their coordinates. It must be applied to the quadratic produced by substitution, with strict or non-strict parameter boundaries matching the requested geometric case.

Explanation

Tangency, distance from a centre and discriminants

A line is tangent to a circle when it has exactly one common point. Algebraically, substitution produces a quadratic with discriminant zero. A positive discriminant gives two intersections and a negative discriminant gives none.

There is also a geometric test: a line is tangent exactly when the perpendicular distance from the circle's centre to the line equals the radius. For ax + by + c = 0, the distance from (x0, y0) is |ax0 + by0 + c|/sqrt(a^2 + b^2).

Absolute values often create two parameter values, representing parallel tangents on opposite sides of a circle. State both unless another condition excludes one, and distinguish equality for tangency from strict inequalities for intersecting or missing the circle.

Explanation

Common errors and exact-answer discipline

Keep subtraction order consistent in a gradient, and do not assign a numerical gradient to a vertical line. When using -a/b from ax + by + c = 0, first confirm b is non-zero. For perpendicular lines, use the negative reciprocal or the horizontal-vertical special case.

In circle equations, read centre signs oppositely from the brackets and remember that the right side is radius squared, not radius. When completing squares, add the same correction to both sides or account for it explicitly before identifying the radius.

For intersections, recover paired coordinates and check every point in both equations. Preserve exact roots unless rounding is requested, state parameter boundaries carefully, and never assume a diagram supplies a point, orientation or intersection that the text has not stated.

Revision checklist and excluded scope

Lines: calculate gradients with matched subtraction order; choose y = mx + c, point-gradient or general form appropriately; and handle vertical lines separately. Use equal gradients for parallel lines and negative reciprocals for non-vertical perpendicular lines.

Coordinates: average coordinates for a midpoint, use Pythagoras for distance, and solve equations simultaneously for intersections. Verify an intersection in every original equation and retain exact values when no rounding is requested.

Circles: identify centre and radius from standard form, complete both squares in expanded form, and use diameter, symmetry, tangent-radius perpendicularity and angle-in-a-semicircle facts only when their conditions are stated.

Intersections: substitute a line into a circle or curve and interpret the quadratic discriminant. Excluded scope includes implicit differentiation, all other Pure Mathematics 1 sections, other examination components and complete AS Mathematics coverage.