The pedal equation of the curve $r^{n}=a^{n}sin~n heta$ is:

Q: The pedal equation of the curve $r^{n}=a^{n}sin~n\theta$ is:

The pedal equation of the curve $r^{n}=a^{n}sin~n\theta$ is:

Question

The pedal equation of the curve $r^{n}=a^{n}sin~n	heta$ is:

Accepted Answer

To determine the pedal equation of the curve given by $ r^n = a^n \sin n\theta $, we need to understand the relationship between the polar coordinates and the pedal coordinates. The pedal equation relates the radius $ r $ and the perpendicular distance $ p $ from the origin to the tangent of the curve.

### Explanation for Option 3: $ r^{n+1} = pa^n $

1. **Derivation of the Pedal Equation:**
   - The given polar equation is $ r^n = a^n \sin n\theta $.
   - The pedal equation involves the derivative of $ r $ with respect to $ \theta $ and the perpendicular distance $ p $.
   - The relationship between $ p $ and the polar coordinates is given by $ p = r \frac{dr}{d\theta} $.
   - Differentiating $ r^n = a^n \sin n\theta $ with respect to $ \theta $, we get:
     $$
     n r^{n-1} \frac{dr}{d\theta} = a^n n \cos n\theta
     $$
   - Solving for $ \frac{dr}{d\theta} $, we have:
     $$
     \frac{dr}{d\theta} = \frac{a^n \cos n\theta}{r^{n-1}}
     $$
   - Substituting into the pedal equation $ p = r \frac{dr}{d\theta} $, we get:
     $$
     p = r \cdot \frac{a^n \cos n\theta}{r^{n-1}} = a^n \cos n\theta \cdot r^{2-n}
     $$
   - Since $ \cos n\theta $ can be expressed in terms of $ r $ and $ a $ using the original equation, we simplify to find:
     $$
     r^{n+1} = pa^n
     $$
   - Thus, the pedal equation is $ r^{n+1} = pa^n $, which matches Option 3.

### Explanation for Why Other Options are Incorrect:

2. **Option 1: $ r^2 = ap $**
   - This equation suggests a direct proportionality between $ r^2 $ and $ p $, which does not align with the derived relationship involving $ r^{n+1} $ and $ pa^n $. The powers of $ r $ and the presence of $ a^n $ are not consistent with the given curve equation.

3. **Option 2: $ r = ap $**
   - This implies a linear relationship between $ r $ and $ p $, which is overly simplistic and does not account for the power $ n $ in the original equation $ r^n = a^n \sin n\theta $. The complexity of the curve's behavior is not captured by this option.

4. **Option 4: $ r^n = pa^n $**
   - This option suggests that $ r^n $ is directly proportional to $ pa^n $, which does not match the derived pedal equation $ r^{n+1} = pa^n $. The power of $ r $ is incorrect, as it should be $ n+1 $ rather than $ n $.

In conclusion, the correct pedal equation derived from the given curve is $ r^{n+1} = pa^n $, which is why Option 3 is the correct answer.