A-level · Beta

Parametric curves

Trace an ellipse and connect its parameter, velocity and tangent.

Drag T around the ellipse. A and B resize its horizontal and vertical radii.

x=3\cos t,\quad y=2\sin t

Tab to a ring or outcome, then use the arrow keys. Sliders also work with touch and keyboard.

The green tangent follows the direction of (dx/dt, dy/dt).

Derivatives use radians. On an ellipse, t usually differs from the point’s angle from the origin. The axes use equal scales.

Predict, test and explain

Step 1 of 3

Predict

For x = 3cos t and y = 2sin t, what is the tangent at t = 0?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

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Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

For x = 3cos t and y = 2sin t, what is the tangent at t = 0?

$x=3\cos t,\quad y=2\sin t,\quad t=0$

Hint

Differentiate x and y with respect to t in radians.

Reveal answer

dx/dt = 0 but dy/dt = 2, so the point moves vertically at (3, 0).

$x=3$

Hint 1

Differentiate x and y with respect to t in radians.

Hint 2

Where dx/dt is nonzero, divide dy/dt by dx/dt.

Hint 3

If dx/dt = 0 and dy/dt ≠ 0, the tangent is vertical.

Worked example

This example uses fixed values, separate from the diagram controls.

Find the point and tangent when t = 0.

Answer: Point (3, 0); tangent x = 3.

Connect this idea

Watch out: An undefined tangent gradient does not mean a horizontal tangent.

Specification and learning route

AQA C3, G5 · Edexcel Pure 3.3, Pure 7.5

A-level extension. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

Useful starting knowledge: Cartesian graphs; Eliminating variables; Functions; Trigonometry for trig parameters.