Lines Meeting Circles: Tangents and Chords
Find where a line cuts a circle by substitution, use the discriminant to tell a tangent from a chord or a miss, and use the right angle between a tangent and the radius.
What a learner can do afterwards
- Find the two intersection points of a line and a circle by substituting
- Show a line is a tangent by getting a discriminant of zero
- Find the tangent at a given point on a circle using the radius gradient
1 · Read
Last stop you could write any circle down and read it back. Now put a straight line across it. On the Ridgeway plan a pipe run is drawn over the round drainage tank, and the crew needs to know whether it crosses the tank, just grazes the wall, or misses it. A line that crosses at two points cuts a chord between them. A line that meets the wall once and turns away is a tangent. The way to tell is not to look at the drawing: substitute the line into the circle equation and count the answers.
Substituting always leaves one quadratic in one variable, and its roots are the x values of the meeting points. So the discriminant b² - 4ac decides the whole question without solving anything. A positive discriminant gives two roots, so the line cuts the circle twice and there is a chord. A discriminant of zero gives one repeated root, so the line touches at a single point and is a tangent. A negative discriminant gives no real roots, so the line misses. Three cases, and the sign of one number picks between them.
Find where y = x + 1 meets x² + y² = 25. Substitute the line for y: x² + (x + 1)² = 25. Expand the bracket carefully: x² + x² + 2x + 1 = 25, so 2x² + 2x - 24 = 0. Divide by 2 to get x² + x - 12 = 0, which factorises as (x + 4)(x - 3) = 0. So x = -4 or x = 3. Put each back into the line, not the circle, because the line is easier: y = -3 and y = 4. The points are (-4, -3) and (3, 4).
Now a tangent test. Is y = 2x + 5 a tangent to x² + y² = 5? Substitute: x² + (2x + 5)² = 5, which expands to x² + 4x² + 20x + 25 = 5, so 5x² + 20x + 20 = 0. Divide by 5: x² + 4x + 4 = 0. The discriminant is 16 - 16 = 0, so there is one repeated root and the line is a tangent. Solving gives x = -2, and the line gives y = 1, so it touches at (-2, 1). A single point of contact is what the zero was promising.
A second route to a tangent needs no quadratic. A tangent meets the radius at that point at a right angle, so their gradients multiply to -1. Take (x - 2)² + (y - 1)² = 25 and the point (5, 5) on it. The radius runs from (2, 1) to (5, 5), so its gradient is (5 - 1) / (5 - 2) = 4/3. The tangent gradient is therefore -3/4. Then y - 5 = -(3/4)(x - 5) tidies to 3x + 4y - 35 = 0. A horizontal radius has no gradient to flip: its tangent is vertical, one radius left or right of the centre.
To see how a line meets a circle, substitute the line into the circle equation and read the quadratic that falls out. A positive discriminant means two points and a chord, zero means one point and a tangent, and a negative value means a miss. Solving the quadratic gives the x values, and the line gives each y. When the point of contact is already known, the tangent can be built instead from the radius gradient, flipped and sign-changed.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
24 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.