Equations of straight lines, gradients, parallel and perpendicular lines, the equation of a circle, tangents and chords, and parametric equations of curves.

What this dot point is asking

AQA wants you to find and use the equation of a straight line, apply the gradient conditions for parallel and perpendicular lines, work with the equation of a circle (centre and radius, tangents, chords and the angle in a semicircle), and use parametric equations to describe curves. Circle questions in particular reward combining several standard facts in one problem.

Straight lines

The gradient between two points is $m = \dfrac{y_2 - y_1}{x_2 - x_1}$. The line through $(x_1, y_1)$ with gradient $m$ is $y - y_1 = m(x - x_1)$, often rearranged to $y = mx + c$ or to the form $ax + by + c = 0$ when the question specifies it. The midpoint of two points is $\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)$, and the distance between them is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

The equation of a circle

Tangents, chords and circle properties

Three circle facts recur in coordinate problems: a tangent is perpendicular to the radius at the point of contact; the perpendicular from the centre to a chord bisects the chord; and the angle in a semicircle is a right angle. To find a tangent at a point, compute the gradient of the radius to that point, take the negative reciprocal for the tangent gradient, and use the point to write the line. To test whether a point lies inside, on, or outside a circle, substitute it into $(x - a)^2 + (y - b)^2$ and compare with $r^2$.

Parametric equations

A curve can be described by $x = f(t)$ and $y = g(t)$, where $t$ is a parameter. To convert to Cartesian form, eliminate $t$, usually by making $t$ the subject of one equation and substituting into the other, or by using an identity (such as $\cos^2 t + \sin^2 t = 1$) when the parameter is an angle. For example, with $x = 2t$ and $y = t^2$, $t = \dfrac{x}{2}$ gives $y = \dfrac{x^2}{4}$, a parabola.

Intersections and a worked strategy for circle problems

Finding where a line meets a circle reduces to substituting the line into the circle equation, giving a quadratic whose discriminant tells you the geometry: two roots mean the line is a secant (two intersection points), one repeated root means it is a tangent (touching once), and no real roots mean it misses the circle. This discriminant test is a frequent exam route to proving a line is a tangent without using gradients.

For circle problems generally, a dependable order of attack is: write the circle in standard form by completing the square to extract the centre and radius; then use whichever circle property the question hints at. A tangent question uses radius-tangent perpendicularity; a chord question uses the perpendicular from the centre bisecting the chord (so the perpendicular distance from the centre, the half-chord, and the radius form a right-angled triangle); and a question mentioning a diameter often uses the angle in a semicircle being a right angle. Identifying which property applies is usually the key step, after which the algebra is routine.