Question

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

Answer 1

Determine line is tangent to a circle:

It depends on what information is given in the problem.

• Either you'll use the knowledge,
• That the radius from the center to $P$ is perpendicular to a line tangent to a circle at point $P$.
• or you'll use the information that the perpendicular meets the circle precisely once.

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

• The following will work if you are given the circle's center (designate it $C$),
• A point on the circle (designate it $P$), and
• A point on the line you are being asked about (designate it $Q$).
• Use the fact that two lines are only perpendicular to one another in the coordinate plane.
• If and only if their slopes are the reciprocals of each other's negative values.
• (or one line is horizontal and the other is vertical).
• Therefore, determine the slopes of $CP$ and $PQ$.
• The lines are perpendicular and $PQ$ is tangent to the circle.
• if they are the negative reciprocals of one another.
• If not, it is not tangential.
• You'll need a different approach if this isn't the information you've been given.
• By locating all sites where the circle and the line meet, can determine (and show) whether the line is (or is not) tangent to the circle
• Given the equations of the circle and the line.
• Tangent indicates one intersection.
• None of more than one, not tangent.

Hence, if one intersection means tangent, none or more than one is not tangent.