Loci with conics

Method

1. Write the moving point M as (x, y) in terms of the parameter t (or other variables).

2. Make cos t and sin t (or cosh t and sinh t, or sec t and tan t) the subjects.

3. Eliminate t with the matching identity: cos² t + sin² t = 1, cosh² t - sinh² t = 1, or sec² t - tan² t = 1.

4. Simplify and name the curve if you can.

Common set-ups

• M is the midpoint of OP, or of P and a fixed point.
• M is where a tangent or normal meets an axis, or the midpoint of the two intercepts.
• M is where two tangents meet.
• M moves so that its distance from a point is e times its distance from a line (a focus-directrix locus).

Example: perpendicular tangents

Tangents y = mx + c to x²/a² + y²/b² = 1 through (h, k) satisfy (k - mh)² = a^2m2 + b², so (h² - a²)m² - 2hkm + k² - b² = 0. If the tangents are perpendicular, the product of the roots is -1: (k² - b²)/(h² - a²) = -1, so h² + k² = a² + b². The locus is a circle of radius sqrt(a² + b²), the director circle.

Worked example

The tangent at P(4 cos t, 3 sin t) to x²/16 + y²/9 = 1 meets the axes at A and B. Find the locus of the midpoint M of AB.

1. Tangent: 3x cos t + 4y sin t = 12, so A = (4/cos t, 0) and B = (0, 3/sin t)
2. M = (2/cos t, 3/(2 sin t)), so cos t = 2/x and sin t = 3/(2y)
3. cos² t + sin² t = 1

Answer: 4/x² + 9/(4y²) = 1

Check yourself

1. P moves on x²/a² + y²/b² = 1. What is the locus of the midpoint of OP?
   • x²/a² + y²/b² = 4
   • x²/(4a²) + y²/(4b²) = 1
   • 4x²/a² + 4y²/b² = 1
   • 2x²/a² + 2y²/b² = 1
   Show answer

   4x²/a² + 4y²/b² = 1. M = ((a cos t)/2, (b sin t)/2), so cos t = 2x/a and sin t = 2y/b. Then 4x²/a² + 4y²/b² = 1: a smaller ellipse.

2. Eliminate t from x = 3 cosh t, y = 3 sinh t.
   • x² - y² = 9
   • x² + y² = 9
   • x² - y² = 3
   • y² - x² = 9
   Show answer

   x² - y² = 9. cosh t = x/3 and sinh t = y/3, and cosh² t - sinh² t = 1 gives x²/9 - y²/9 = 1.

3. A point moves with x = 1 + 2 cos t, y = sin t. Its locus is an ellipse. Find the x-coordinate of the centre.
   Show answer

   1. cos t = (x - 1)/2 and sin t = y, so (x - 1)²/4 + y² = 1, an ellipse centred at (1, 0).

4. A point P moves so that its distance from (4, 0) is half its distance from the line x = 16. Its locus is x²/a² + y²/b² = 1. Find a².
   Show answer

   64. e = 1/2 and the focus is (ae, 0) = (4, 0), so a = 8. Check: the directrix is x = a/e = 16. So a² = 64 (and b² = 64(1 - 1/4) = 48).

5. Find the radius of the circle on which perpendicular tangents to x²/16 + y²/9 = 1 meet.
   Show answer

   5. The locus is x² + y² = a² + b² = 25, a circle of radius 5.

6. A point moves with x = 1 + 2 sec t, y = 3 tan t. Its locus is a hyperbola. Find the x-coordinate of its right-hand vertex.
   Show answer

   3. sec t = (x - 1)/2 and tan t = y/3, so (x - 1)²/4 - y²/9 = 1. The centre is (1, 0) and a = 2, so the vertex is at x = 3.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics, Further Mathematics and Pure Mathematics specification (issue 3), Unit FP3

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