Ellipses, velocities and tangent gradients.
Example 1
An ellipse is parameterised by x=12cos(t),y=3sin(t). Find dxdy at t=4π.
Explore the curve. The gold ring marks the position in the question.Show worked answer
Answer: 12−3
- dxdy=−12sin(t)3cos(t).
- At 4π, sine and cosine are equal and non-zero, giving 12−3.
Example 2
A particle moves with x=12cos(t),y=3sin(t). Find its speed at t=0.
Explore the curve. The gold ring marks the position in the question.Show worked answer
Answer: 3
- Velocity is (−12sin(t),3cos(t)).
- At t=0 this is (0,3), whose magnitude is 3.
Example 3
An ellipse is parameterised by x=2cos(t),y=3sin(t). Find dxdy at t=4π.
Explore the curve. The gold ring marks the position in the question.Show worked answer
Answer: 2−3
- dxdy=−2sin(t)3cos(t).
- At 4π, sine and cosine are equal and non-zero, giving 2−3.