Home > Graphing > Lines intersecting circles

## Lines intersecting circles

When a line intersects a circle it will either have one point of contact in which case it is a tangent to the circle or it will cut it in two places.

To find the points of contact between a line and a circle

1. If needed, rearange the equation of the line to make x or y the subject
2. Substitute into the circle equation and solve the resulting quadratic
3. Substitute the solutions into the equation of the line to get the required coordinates

Example

Find where the line $y=2x+1$ intersects with the circle $(x+2)^2+(y-1)^2=5$

When we substitute y into the circle we get $(x+2)^2 + ((2x+1)-1)^2=5$

$\therefore (x+2)^2+(2x)^2=5 \Rightarrow x^2+4x+4+4x^2=5 \Rightarrow 5x^2+4x-1=0$

$\therefore (5x-1)(x+1)=0 \Rightarrow x=\frac{1}{5}$ or $x = -1$

If $x=\frac{1}{5}, y=\frac{7}{5}$

If $x=-1, y=-1$

$\therefore$ the points of intersection are $(\frac{1}{5}, \frac{7}{5})$ and $(-1,-1)$