Show that the straight line with equation is a tangent to the circle with equation .
There are two ways in which a circle and a line might intersect.
They might also not intersect at all. A circle and a line might intersect once or twice.
A singular intersection can only occur if the line is a tangent, so for this problem we need only show that there is one intersection.
The equation for the line can be re-written as
Substituting this for y in the equation of the circle,
Squaring out the bracket and multiplying by nine,
Using the determinant on this quadratic,
This shows there is only one intersection, so the line is a tangent to the circle.