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Tangents and normals for an ellipse Consider the ellipse
Chapter 12, Problem 34RE(choose chapter or problem)
Tangents and normals to an ellipse Consider the ellipse \(\mathbf{r}(t)=\langle 3 \cos t, 4 \sin t\rangle\), for \(0 \leq t \leq 2 \pi\).
(a) Find the tangent vector \(\mathbf{r}^{\prime}\), the unit tangent vector T, and the principal unit normal vector N at all points on the curve.
(b) At what points does \(\left|\mathbf{r}^{\prime}\right|\) have maximum and minimum values?
(c) At what points does the curvature have maximum and minimum values? Interpret this result in light of part (b).
(d) Find the points (if any) at which r and N are parallel.
Questions & Answers
QUESTION:
Tangents and normals to an ellipse Consider the ellipse \(\mathbf{r}(t)=\langle 3 \cos t, 4 \sin t\rangle\), for \(0 \leq t \leq 2 \pi\).
(a) Find the tangent vector \(\mathbf{r}^{\prime}\), the unit tangent vector T, and the principal unit normal vector N at all points on the curve.
(b) At what points does \(\left|\mathbf{r}^{\prime}\right|\) have maximum and minimum values?
(c) At what points does the curvature have maximum and minimum values? Interpret this result in light of part (b).
(d) Find the points (if any) at which r and N are parallel.
ANSWER:Solution 34RE
Step 1:
The tangent vector
Here
So,
Unit tangent vector :
For normal vector, we need the derivative of tangent vector and its magnitude.
Normal vector is
b.