In the curve `x^(m+n)=a^(m-n)y^(2n)`, mth power of subtangent varies as the nth power of the sub-normal.

Answer

To prove that the mth power of the subtangent varies as the nth power of the sub-normal for the curve given by the equation \( x^{m+n} = a^{m-n} y^{2n} \), we will follow these steps:

### Step 1: Differentiate the given equation
We start with the equation of the curve:
\[ x^{m+n} = a^{m-n} y^{2n} \]
Differentiating both sides with respect to \( x \):
\[ \frac{d}{dx}(x^{m+n}) = \frac{d}{dx}(a^{m-n} y^{2n}) \]
Using the product rule on the right side:
\[ (m+n)x^{m+n-1} = a^{m-n} \cdot 2n y^{2n-1} \frac{dy}{dx} \]

### Step 2: Solve for \(\frac{dy}{dx}\)
Rearranging the equation to solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{(m+n)x^{m+n-1}}{2n a^{m-n} y^{2n-1}} \]

### Step 3: Define subtangent and sub-normal
The subtangent \( T \) and sub-normal \( N \) are defined as follows:
- Subtangent:
\[ T = \left| \frac{y}{\frac{dy}{dx}} \right| \]
- Sub-normal:
\[ N = y \cdot \frac{dy}{dx} \]

### Step 4: Calculate \( T \) and \( N \)
Substituting \(\frac{dy}{dx}\) into the formula for \( T \):
\[ T = \left| \frac{y}{\frac{(m+n)x^{m+n-1}}{2n a^{m-n} y^{2n-1}}} \right| = \frac{2n a^{m-n} y^2}{(m+n)x^{m+n-1}} \]

For \( N \):
\[ N = y \cdot \frac{(m+n)x^{m+n-1}}{2n a^{m-n} y^{2n-1}} = \frac{(m+n)x^{m+n-1} y}{2n a^{m-n} y^{2n-1}} = \frac{(m+n)x^{m+n-1}}{2n a^{m-n} y^{2n-2}} \]

### Step 5: Establish the relationship between \( T^m \) and \( N^n \)
Now we need to find \( T^m \) and \( N^n \):
\[ T^m = \left( \frac{2n a^{m-n} y^2}{(m+n)x^{m+n-1}} \right)^m \]
\[ N^n = \left( \frac{(m+n)x^{m+n-1}}{2n a^{m-n} y^{2n-2}} \right)^n \]

### Step 6: Show that \( T^m \) is proportional to \( N^n \)
From the expressions of \( T^m \) and \( N^n \), we can see that:
\[ T^m \propto N^n \]
This means that the mth power of the subtangent is directly proportional to the nth power of the sub-normal.

### Conclusion
Thus, we have proven that the mth power of the subtangent varies as the nth power of the sub-normal for the given curve.

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