Circle propertiesEdexcel International A Level Maths: Revision notes

Edexcel IAL expects you to use three circle properties together with coordinate geometry (gradients, midpoints, distances and equations of lines):

• the angle in a semicircle is a right angle;
• the perpendicular from the centre to a chord bisects the chord;
• a tangent is perpendicular to the radius at the point of contact. Each one turns a geometric fact into an algebraic condition: two lines are perpendicular when the product of their gradients is $-1$, i.e. $m_1m_2=-1$. The circle equation is $(x-a)^2+(y-b)^2=r^2$, with centre $(a,b)$ and radius $r$.

If $AB$ is a diameter and $P$ is on the circle, then angle $APB=90^\circ$. In coordinates, find the gradients of $AP$ and $BP$; if their product is $-1$ the lines are perpendicular. For $A=(1,2)$, $B=(9,8)$, $P=(8,1)$: $m_{AP}=-\frac17$ and $m_{BP}=7$, product $-1$, so $P$ is on the circle with diameter $AB$. The converse also works: if angle $APB=90^\circ$, then $P$ lies on the circle with diameter $AB$, whose centre is the midpoint of $AB$ and whose radius is $\frac12AB$.

The perpendicular from the centre $C$ to a chord $PQ$ meets $PQ$ at its midpoint $M$. This gives two routes:

• the line $CM$ has gradient $-\frac{1}{m_{PQ}}$ and passes through $C$, so you can find $M$ without solving for $P$ and $Q$;
• triangle $CMQ$ is right-angled at $M$, so $CM^2+MQ^2=r^2$. Worked example: circle centre $(2,3)$, radius $5$, chord $PQ$ with $P=(5,7)$, $Q=(6,0)$. $PQ=\sqrt{50}$, so $MQ^2=12.5$ and $CM^2=25-12.5=12.5$, giving $CM=\frac{5\sqrt2}{2}$.

The tangent at a point $T$ on a circle is perpendicular to the radius $CT$. To find the tangent at $T$:

1. find the gradient $m$ of the radius $CT$;
2. the tangent has gradient $-\frac1m$;
3. use $y-y_1=m(x-x_1)$ with the point $T$. Example: $(x-1)^2+(y+2)^2=25$ at $T=(4,2)$. The radius gradient is $\frac{2-(-2)}{4-1}=\frac43$, so the tangent gradient is $-\frac34$ and $y-2=-\frac34(x-4)$, i.e. $3x+4y-20=0$. If the radius is horizontal the tangent is vertical ($x=k$), and vice versa.

Typical problems ask you to: show that a line meets a circle (substitute into the equation and solve a quadratic); find the midpoint of a chord using the perpendicular from the centre; find lengths with Pythagoras; and find areas of triangles formed by the centre, chord ends or tangent intercepts. Always state the property you use in words (for example 'the angle in a semicircle is $90^\circ$') because a mark is often given for the reason. Give exact values (surds) unless a decimal is requested.