Quick questions on Coordinate geometry: straight lines, circles and parametric curves - OCR A-Level Maths A

A circle with centre $(a, b)$ and radius $r$ has equation $(x - a)^2 + (y - b)^2 = r^2$. When a circle is given in expanded form, complete the square in $x$ and in $y$ to recover the centre and radius.

To find the intersection of a line and a circle, substitute the line into the circle equation and solve the resulting quadratic. The discriminant of that quadratic tells you whether the line is a tangent (one solution, $\Delta = 0$), a secant cutting the circle twice ($\Delta > 0$), or misses it entirely ($\Delta < 0$). This is a favourite OCR synoptic link between coordinate geometry and the discriminant.

If you know three points on a circle, the centre is equidistant from all of them, so it lies on the perpendicular bisectors of the chords joining them. Finding two perpendicular bisectors and solving them simultaneously locates the centre, after which the radius is the distance to any of the three points. This blends the line tools (midpoint, perpendicular gradient) with the circle definition.

$(x + 1)^2$ comes from $x^2 + 2x$ and gives centre $x$-coordinate $-1$ (opposite sign).

Find the equation of the line through $(2, -1)$ perpendicular to $y = 2x + 3$. [3 marks]

State the centre and radius of $(x - 4)^2 + (y + 1)^2 = 16$. [2 marks]