Question

How can I tell if a line is tangent to a circle?

Answer 1

It depends on what information you have.

You'll probably either use the fact that a line tangent to a circle at point `P` is perpendicular to the radius from the center to `P` or you'll use the fact that the perpendicular intersects the circle in exactly one point..

I'm going to guess that you are working in the coordinate plane,

If that you've been given the center of the circle (call it `C`), a point on the circle (`P`) and another point on the line you're being asked about (`Q`), the the following will work.

Use the fact that two lines in the coordinate plane are perpendicular if and only if their slopes are negative reciprocals of each other (or one line is horizontal and the other is vertical).

So find the slope s of `CP` and `PQ`. Is they are negative reciprocals of each other then the lines are perpendicular and `PQ` is tangent to the circle. Otherwise it is not tangent.

If this is not the information you've been given, then you'll need a different method.

Given the equation of the circle and the equation of the line, you can determine (and prove) the line is (or isn't) tangent to the circle by finding all points of intersection of the circle and the line. One intersection means tangent. None or more than one, not tangent.

Answer 2

The easiest way is to solve the system between the equation of the circle and the equation of the line.

The equation that solves that system is quadratic. If the `Delta` of that equation is zero, the line is tangent, if it is positive the line is secant, if it is negative the line is external.

Answer 3

Without getting (explicitly) into slopes if you have an equation for the circle and an equation for the line (2 equations in 2 unknowns) you should be able to solve for points of intersection between the two equations:
If you get exactly one point the line is tangent to the circle.
(zero points implies that the line and circle do not intersect; 2 points imply the line cuts the circle in two places and is therefore not a tangent.)