TMUA Coordinate Geometry: Lines and Circles

Straight lines

Parallel lines have equal gradients. Perpendicular gradients multiply to $-1$. Use $y-y_1=m(x-x_1)$ to build an equation from a point and a gradient.

Circles

$(x-a)^2+(y-b)^2=r^2$ has centre $(a,b)$ and radius $r$. If the equation is $x^2+y^2+cx+dy+e=0$, complete the square in $x$ and in $y$ to read off the centre and radius.

Tangents

A tangent to a circle is perpendicular to the radius at the point of contact, so its gradient is the negative reciprocal of the radius gradient.

Useful facts

The distance between $(x_1,y_1)$ and $(x_2,y_2)$ is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$, and the midpoint averages the coordinates. A perpendicular from the centre of a circle to a chord bisects the chord.

To test whether a point is inside, on or outside a circle, compare its squared distance from the centre with $r^2$.

Worked examples

Common mistakes

• Using the reciprocal of a gradient without changing the sign.
• Reading the centre as $(-a,-b)$ from $(x-a)^2+(y-b)^2$: the signs are the opposite of what is inside the brackets.
• Forgetting that the right-hand side of the circle equation is $r^2$, not $r$.
• Not rearranging the line to $y=mx+c$ before reading its gradient.

Test yourself on this topic with our free practice questions.