Straight lines: the three forms

Three equivalent ways of writing a line — each is convenient in a different situation.

• y = mx + c — gradient m, y-intercept c. Best for reading off a gradient.
• y − y₁ = m(x − x₁) — the workhorse when you know a point and a gradient.
• ax + by + c = 0 — the form AQA often demands for a final answer (integer coefficients).

Parallel and perpendicular

Two straight lines with gradients m₁ and m₂:

• Parallel ⇔ m₁ = m₂.
• Perpendicular ⇔ m₁ m₂ = −1, i.e. m₂ is the negative reciprocal of m₁.

Always rearrange to y = mx + c before reading a gradient. Reading "3" off 3x + 4y = 12 is the single most common error in this topic.

Write the circle

The equation of a circle

A circle is the set of points a fixed distance from a fixed point — which, by Pythagoras, gives:

So (x + 2)² + (y − 5)² = 9 has centre (−2, 5) and radius 3. The signs flip, and the right-hand side is r², not r.

Do not forget to subtract the 9 and the 16 that completing the square introduces — and remember to take the square root at the end.

Parallel, perpendicular or neither?

Tap a pair of lines, then tap the relationship between them. Rearrange to y = mx + c first!

The three circle properties

AQA names three circle facts, and every circle question leans on one of them:

• The angle in a semicircle is a right angle — so if ∠APB = 90°, then AB is a diameter, and the centre is the midpoint of AB.
• The perpendicular from the centre to a chord bisects the chord — so the perpendicular bisector of any chord passes through the centre.
• The tangent is perpendicular to the radius at the point of contact.

Inside, on or outside? Compare the distance from the centre with r. Distance from (3, −4) to the origin is √(3² + 4²) = 5, and 5 < 6, so the origin lies inside this circle.

Tangent and radius

Intersections: line and circle

To find where a line meets a circle, substitute the line into the circle equation. You get a quadratic in x, and its discriminant answers the geometry:

• Δ > 0 — the line is a secant: two intersection points.
• Δ = 0 — the line is a tangent: one repeated point of contact.
• Δ < 0 — the line misses the circle entirely.

Alternative for tangency: the perpendicular distance from the centre to the line equals r. Either method is acceptable — the discriminant route is usually safer in an exam because it needs no formula sheet.

Your turn — a gradient

Your turn — perpendicular gradient

Parametric equations

Instead of linking x and y directly, a parameter t drives both. Each value of t gives one point on the curve.

Domain restrictions survive. If t ≥ 0 then x = 3t means x ≥ 0 — the Cartesian equation alone would wrongly suggest the whole parabola.

Perpendicular line

Your turn — radius of a circle

Trigonometric parameters

When x and y involve sin and cos, eliminate the parameter using the Pythagorean identity rather than by rearranging.

Ellipses too: x = 4cos t, y = 3sin t gives (x/4)² + (y/3)² = 1. The trick is always to isolate cos t and sin t first, then square and add.

Match the question to its answer

Tap an item on the left, then its partner on the right.

Putting it together

Longer AQA questions chain the tools together. A typical route:

• Given the circle in expanded form, complete the square to get the centre and radius.
• Given two points, use the midpoint ((x₁+x₂)/2, (y₁+y₂)/2) and the distance formula √((x₂−x₁)² + (y₂−y₁)²).
• For the perpendicular bisector of a chord: midpoint of the chord, negative reciprocal of the chord gradient — and it must pass through the centre.
• For a tangent length from an external point T: it is √(TC² − r²), by Pythagoras on the right angle between tangent and radius.

Sketch it. A rough diagram catches sign errors and tells you instantly whether an answer is plausible.

Eliminate the parameter

Your turn — inside or outside?

Your turn — parametric point

The big ideas to know

That is AQA section C — Coordinate geometry in the (x, y) plane. Press Finish to see your score.