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The rectangular hyperbola xy = c2

Edexcel IAL Further Mathematics · WFM01 Further Pure Mathematics 1

The rectangular hyperbola xy = c2 is the reciprocal graph y = c2/x. Its asymptotes are the coordinate axes, which meet at right angles: that is why it is called rectangular.

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Cartesian form

xy = c2 (c > 0) has two branches, in the first and third quadrants. The asymptotes are x = 0 and y = 0. It is symmetric in the lines y = x and y = -x, and meets y = x at (c, c) and (-c, -c).

Parametric form

x = ct, y = c/t (t ≠ 0). Check: xy = ct × c/t = c2. Positive t gives the first-quadrant branch, negative t the third.

Example: xy = 9 has c = 3, and the point with t = 2 is (6, 3/2).

Meeting a line

Substitute the line into xy = c2 to get a quadratic. Two roots: the line crosses the curve twice. One repeated root: the line is a tangent. No real roots: they do not meet.

Worked example

Find where the line y = x + 1 meets the curve xy = 12.

  1. x(x + 1) = 12, so x2 + x - 12 = 0.
  2. (x + 4)(x - 3) = 0, so x = 3 or x = -4.
  3. y = x + 1 gives y = 4 or y = -3.

Answer: (3, 4) and (-4, -3)

Key idea: xy = c2: parametric (ct, c/t); asymptotes are the axes; symmetric about y = ±x.

Check yourself

  1. The rectangular hyperbola xy = 16 has parametric equations x = ct, y = c/t, with c > 0. Find c.
    Show answer

    4. c2 = 16, so c = 4.

  2. Find the y-coordinate of the point with parameter t = 2 on xy = 9. Give your answer as a fraction in its simplest form.
    Show answer

    3/2. c = 3, so the point is (ct, c/t) = (6, 3/2).

  3. Find the x-coordinates of the points where y = x + 1 meets xy = 12.
    Show answer

    3, -4. x(x + 1) = 12 gives (x + 4)(x - 3) = 0.

  4. What are the asymptotes of xy = c2?
    • y = x and y = -x
    • x = c and y = c
    • x = c2 and y = c2
    • x = 0 and y = 0
    Show answer

    x = 0 and y = 0. As x grows, y = c2/x tends to 0; as x tends to 0, y grows without bound.

  5. A curve has parametric equations x = 3t, y = 3/t. Its Cartesian equation is xy = k. Find k.
    Show answer

    9. xy = 3t × 3/t = 9.

  6. Find the x-coordinates of the points where y = 4x meets xy = 36.
    Show answer

    3, -3. x(4x) = 36 gives x2 = 9.

  7. The curve xy = 25 meets the line y = x at (p, p) with p > 0. Find p.
    Show answer

    5. x2 = 25 gives x = 5 (the first-quadrant point), which is c.

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