All subjects · Further Mathematics · WFM03 Further Pure Mathematics 3

When is a line a tangent?

Edexcel IAL Further Mathematics · WFM03 Further Pure Mathematics 3

A line touches a conic when substituting it in gives a quadratic with one repeated root. This gives a neat condition linking m and c, and lets you find tangents with a given gradient or through a given point.

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The discriminant method

Substitute y = mx + c into the curve. You get a quadratic in x.

The conditions

Ellipse x2/a2 + y2/b2 = 1: y = mx + c is a tangent if c2 = a2m2 + b2.

Hyperbola x2/a2 - y2/b2 = 1: y = mx + c is a tangent if c2 = a2m2 - b2.

Be ready to derive these: substitute, collect terms, and set the discriminant to zero.

Tangents through a given point

If the tangent must pass through (h, k), then k = mh + c. Substitute c = k - mh into the tangency condition to get a quadratic in m. Its two roots are the gradients of the two tangents from that point.

Worked example

Show that y = mx + c is a tangent to x2/a2 + y2/b2 = 1 when c2 = a2m2 + b2.

  1. Substitute: b2x2 + a2(mx + c)2 = a2b2
  2. (b2 + a2m2)x2 + 2a2mcx + a2(c2 - b2) = 0
  3. Discriminant zero: 4a4m2c2 - 4(b2 + a2m2)a2(c2 - b2) = 0
  4. Divide by 4a2b2 and simplify: -c2 + b2 + a2m2 = 0

Answer: c2 = a2m2 + b2

Key idea: Substitute and set the discriminant to zero. Ellipse: c2 = a2m2 + b2. Hyperbola: c2 = a2m2 - b2.

Check yourself

  1. y = 2x + c, with c > 0, is a tangent to x2/4 + y2/9 = 1. Find c.
    Show answer

    5. c2 = a2m2 + b2 = 4(4) + 9 = 25, so c = 5.

  2. y = mx + 3, with m > 0, is a tangent to x2/8 + y2 = 1. Find m.
    Show answer

    1. 9 = 8m2 + 1, so m2 = 1 and m = 1.

  3. y = x + c, with c > 0, is a tangent to x2/16 - y2/9 = 1. Find c, to 3 decimal places.
    Show answer

    2.646. c2 = a2m2 - b2 = 16 - 9 = 7, so c = sqrt(7) = 2.646.

  4. After substituting y = mx + c into a conic, what tells you the line is a tangent?
    • The quadratic in x has discriminant zero
    • The quadratic in x has two positive roots
    • The constant term is zero
    • The coefficient of x2 is zero
    Show answer

    The quadratic in x has discriminant zero. A tangent meets the curve at one repeated point.

  5. What happens to the line y = 3x + 1 and the hyperbola x2 - y2/4 = 1?
    • The line is a tangent
    • They meet at two points
    • They do not meet
    • The line is an asymptote
    Show answer

    They do not meet. Substituting gives 4x2 - (3x + 1)2 = 4, so -5x2 - 6x - 5 = 0, with discriminant 36 - 100 = -64 < 0.

  6. Tangents to x2/9 + y2/4 = 1 pass through (0, 4). Find the positive gradient, to 3 decimal places.
    Show answer

    1.155. y = mx + 4, so c = 4. 16 = 9m2 + 4 gives m2 = 4/3 and m = 2/sqrt(3) = 1.155.

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Last updated 8 November 2026. Spotted a mistake? Tell us.