Find the equation of the normal to the curve `ay^(2)=x^(3)` at the point `(am^(2), am^(3))`.

Answer

To find the equation of the normal to the curve \( ay^2 = x^3 \) at the point \( (am^2, am^3) \), we will follow these steps:

### Step 1: Differentiate the curve
We start with the equation of the curve:
\[ ay^2 = x^3 \]
To find the slope of the tangent, we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(ay^2) = \frac{d}{dx}(x^3) \]
Using the chain rule on the left side, we get:
\[ 2ay \frac{dy}{dx} = 3x^2 \]

### Step 2: Solve for \(\frac{dy}{dx}\)
Rearranging the equation to solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{3x^2}{2ay} \]

### Step 3: Substitute the point into the derivative
Now, we substitute the point \( (am^2, am^3) \) into the derivative:
\[ x = am^2 \quad \text{and} \quad y = am^3 \]
Substituting these values into the derivative:
\[ \frac{dy}{dx} = \frac{3(am^2)^2}{2a(am^3)} = \frac{3a^2m^4}{2a^2m^3} = \frac{3m}{2} \]

### Step 4: Find the slope of the normal
The slope of the tangent line at the point is \( \frac{3m}{2} \). The slope of the normal line is the negative reciprocal of the slope of the tangent:
\[ \text{slope of normal} = -\frac{1}{\frac{3m}{2}} = -\frac{2}{3m} \]

### Step 5: Use point-slope form to find the equation of the normal
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
where \( m \) is the slope of the normal, and \( (x_1, y_1) = (am^2, am^3) \):
\[ y - am^3 = -\frac{2}{3m}(x - am^2) \]

### Step 6: Simplify the equation
Expanding this equation:
\[ y - am^3 = -\frac{2}{3m}x + \frac{2}{3m}(am^2) \]
Rearranging gives:
\[ y = -\frac{2}{3m}x + \frac{2a m}{3} + am^3 \]

### Step 7: Combine terms
To combine the constant terms:
\[ y = -\frac{2}{3m}x + \left(\frac{2a m + 3a m^3}{3}\right) \]
Thus, the equation of the normal line can be written as:
\[ 2x + 3my - (2am + 3am^3) = 0 \]

### Final Answer
The equation of the normal to the curve \( ay^2 = x^3 \) at the point \( (am^2, am^3) \) is:
\[ 2x + 3my - (2am + 3am^3) = 0 \]