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To find the evolute of the given ellipse equation:

\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,
\]

we can proceed as follows:

### Step-by-step process:

1. **Parametric Form of the Ellipse:**
   The ellipse can be parameterized as:
   \[
   x = a \cos \theta, \quad y = b \sin \theta,
   \]
   where \( \theta \) is the parameter.

2. **First and Second Derivatives of Parametric Equations:**
   For the parameterized coordinates, we calculate the first derivatives with respect to \( \theta \):
   \[
   \frac{dx}{d\theta} = -a \sin \theta, \quad \frac{dy}{d\theta} = b \cos \theta.
   \]
   
   The second derivatives are:
   \[
   \frac{d^2x}{d\theta^2} = -a \cos \theta, \quad \frac{d^2y}{d\theta^2} = -b \sin \theta.
   \]

3. **Radius of Curvature (\( R \)):**
   The formula for the radius of curvature \( R \) in parametric form is:
   \[
   R = \frac{\left( \left( \frac{dx}{d\theta} \right)^2 + \left( \frac{dy}{d\theta} \right)^2 \right)^{3/2}}{\left| \frac{dx}{d\theta} \frac{d^2y}{d\theta^2} - \frac{dy}{d\theta} \frac{d^2x}{d\theta^2} \right|}.
   \]
   
   Plugging in the values of the first and second derivatives, we get:
   \[
   R = \frac{\left( a^2 \sin^2 \theta + b^2 \cos^2 \theta \right)^{3/2}}{ab}.
   \]

4. **Center of Curvature Coordinates:**
   The coordinates of the center of curvature \( (x_c, y_c) \) for an ellipse are given by:
   \[
   x_c = x - \frac{(b^2/a) \cos^3 \theta}{(a^2 \sin^2 \theta + b^2 \cos^2 \theta)^{1/2}},
   \]
   \[
   y_c = y - \frac{(a^2/b) \sin^3 \theta}{(a^2 \sin^2 \theta + b^2 \cos^2 \theta)^{1/2}}.
   \]

   Thus, the parametric equations for the evolute of the ellipse describe the path traced by the center of curvature as the point moves along the ellipse.

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Would you like further details or have any questions on this solution? Here are some related questions to explore:

1. What is the significance of an evolute in geometry?
2. How do the parameters \(a\) and \(b\) affect the shape of the evolute?
3. Can we find the evolute for a circle using a similar method?
4. What is the geometric interpretation of the radius of curvature?
5. How does the curvature vary along different points of an ellipse?

**Tip:** The evolute is the envelope of normals to the curve, meaning it's a curve where all the centers of curvature lie.

Find the evolute of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Explore the detailed solution to finding the evolute of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) using parametric equations and curvature formulas. Ideal for students studying undergraduate calculus.

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