The length of chord of curvature through the pole of the equiangular spiral $r=ae^{m heta}$ is

Question

The length of chord of curvature through the pole of the equiangular spiral $r=ae^{m	heta}$ is

Accepted Answer

To determine the length of the chord of curvature through the pole of the equiangular spiral given by the equation $ r = ae^{m\theta} $, we need to understand the properties of the equiangular spiral and the concept of the chord of curvature.

### Explanation:

1. **Equiangular Spiral Properties:**
   - An equiangular spiral, also known as a logarithmic spiral, is defined by the polar equation $ r = ae^{m\theta} $.
   - It has the unique property that the angle between the tangent and radial line at any point is constant.

2. **Chord of Curvature:**
   - The chord of curvature through a point on a curve is a line segment that passes through the point and is perpendicular to the radius of curvature at that point.
   - For polar curves, the length of the chord of curvature through the pole can be calculated using specific formulas derived from the geometry of the curve.

3. **Calculation for Equiangular Spiral:**
   - For the equiangular spiral $ r = ae^{m\theta} $, the radius of curvature $ \rho $ at any point is given by:
     $$
     \rho = \frac{r \sqrt{1 + m^2}}{m}
     $$
   - The chord of curvature through the pole is twice the radius of curvature at the pole.
   - At the pole, the radius $ r $ is simply the value of the spiral at that point, which is $ r = ae^{m\theta} $.

4. **Length of Chord of Curvature:**
   - The length of the chord of curvature through the pole is calculated as:
     $$
     \text{Chord length} = 2 \times \rho = 2 \times \frac{r \sqrt{1 + m^2}}{m}
     $$
   - However, due to the specific nature of the equiangular spiral, the chord length simplifies to $ 2r $ when considering the geometry and the properties of the spiral.

### Justification for Option 1:

- **Option 1: 2r** is correct because the length of the chord of curvature through the pole for an equiangular spiral is indeed twice the radial distance at the pole, which simplifies to $ 2r $.

### Why Other Options are Incorrect:

- **Option 2: r** is incorrect because the chord of curvature is not simply the radial distance; it is twice the radial distance due to the properties of the equiangular spiral.
- **Option 3: 4r** is incorrect because it overestimates the length of the chord of curvature. The correct factor is 2, not 4.
- **Option 4: 3r** is incorrect because it does not align with the geometric properties of the equiangular spiral, which dictate that the chord length is $ 2r $.

In conclusion, the correct answer is Option 1: 2r, as it accurately reflects the length of the chord of curvature through the pole for the given equiangular spiral.