How do you find the equation of a line at a given point that is tangent to a circle?

This question may start by giving you an equation, which you need to write in the form of the standard equation of a circle (usually by completing the square). This allows you to see the centre of the circle. As you now have two coordinates you can work out the gradient of the line between the centre of the circle and the given point on the circumference. As this line is perpendicular to the tangent on the circle, the gradient is the negative reciprocal. Now you know the gradient of the tangent and a point it crosses through, you can find the equation of the line by substituting these values into the standard equation of a line y=mx+c