--- title: "Lines Meeting Circles: Tangents and Chords" description: "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." canonical: https://lightmysky.com/learn/mathematics/lines-meeting-circles-tangents-and-chords-mt_ELA4IzIVdv source: https://lightmysky.com/learn/mathematics/lines-meeting-circles-tangents-and-chords-mt_ELA4IzIVdv.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Mathematics · Area: Geometry · Ages 16 to 17 Page: https://lightmysky.com/learn/mathematics/lines-meeting-circles-tangents-and-chords-mt_ELA4IzIVdv ## Ready when they can - 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 ## Lesson: Lines Meeting Circles: Tangents and Chords 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. *(drawing: One number, three verdicts. Work it out before solving anything.)* **Example.** 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). **Example.** 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. *(drawing: Two equations become one quadratic, and the quadratic answers the question.)* **Example.** 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. **Recap.** 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. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [The Equation of a Circle](https://lightmysky.com/learn/mathematics/the-equation-of-a-circle-mt_YARqmi_ioe) - [The Discriminant and the Number of Roots](https://lightmysky.com/learn/mathematics/the-discriminant-and-the-number-of-roots-mt_uj95V0aLVS) ## Opens up - [The Gradient of a Curve as a Limit](https://lightmysky.com/learn/mathematics/the-gradient-of-a-curve-as-a-limit-mt_LAlhcwVjln)