How do you find the centre and radius from a general-form equation?

Complete the square in x and in y, then read off the centre and r^2.

How do you tell if a point is inside, on or outside a circle?

Compare its distance^2 from the centre with r^2: smaller, equal or larger.

Angle in a semicircle?

A right angle: if AB is a diameter and P is on the circle then AP\perp BP.

Perpendicular from the centre to a chord?

It bisects the chord, so the perpendicular bisector of a chord passes through the centre.

Method for the tangent at a point A on a circle?

Gradient of radius CA, negative reciprocal, then y-y1=m(x-x1).

How do you find the centre when given the endpoints of a diameter?

It is the midpoint of the two endpoints.

How do you find the circumcircle of a triangle?

Intersect two perpendicular bisectors of sides for the centre, then use the distance to a vertex as the radius.

CirclesEdexcel A-Level Maths: Flashcards

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Equation of a circle with centre $(a,b)$ and radius $r$?

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Equation of a circle with centre $(a,b)$ and radius $r$ ?: $(x-a)^2+(y-b)^2=r^2$
Centre and radius of $x^2+y^2+2fx+2gy+c=0$ ?: Centre $(-f,-g)$ ; $r^2=f^2+g^2-c$
How do you find the centre and radius from a general-form equation?: Complete the square in $x$ and in $y$ , then read off the centre and $r^2$ .
How do you tell if a point is inside, on or outside a circle?: Compare its distance $^2$ from the centre with $r^2$ : smaller, equal or larger.
Angle in a semicircle?: A right angle: if $AB$ is a diameter and $P$ is on the circle then $AP\perp BP$ .
Perpendicular from the centre to a chord?: It bisects the chord, so the perpendicular bisector of a chord passes through the centre.
Relationship between a tangent and the radius at the point of contact?: They are perpendicular.
Gradient of a line perpendicular to one with gradient $m$ ?: $-\frac1m$ (so $m_1m_2=-1$ )
Method for the tangent at a point $A$ on a circle?: Gradient of radius $CA$ , negative reciprocal, then $y-y_1=m(x-x_1)$ .
Where does the normal to a circle pass through?: The centre of the circle.
How do you find the centre when given the endpoints of a diameter?: It is the midpoint of the two endpoints.
How do you find the circumcircle of a triangle?: Intersect two perpendicular bisectors of sides for the centre, then use the distance to a vertex as the radius.

Exam questions on Circles

A circle $C$ has equation $x^2+y^2-6x+4y-12=0$ .
Determine whether the point $(6,3)$ lies inside, outside or on $C$ . Show your working.2 marks
A circle has centre $C(2,1)$ and passes through the point $A(5,5)$ .
Find the equation of the tangent to the circle at $A$ , giving your answer in the form $ax+by+c=0$ , where $a$ , $b$ and $c$ are integers.2 marks
The points $A(-1,2)$ and $B(5,10)$ are the endpoints of a diameter of a circle $S$ .
Find the equation of $S$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).