If ` x^(4)y^(5) =(x+y) ^(m+1) and (dy)/(dx) =(y)/(x) ,then m=`

Options

A. 8
B. 4
C. 5
D. 20

Answer

To solve the problem, we start with the given equation and the derivative condition.

### Step-by-Step Solution:

1. **Given Equation**:
\[ x^4 y^5 = (x + y)^{m + 1} \]

2. **Differentiate Both Sides**:
We will use the product rule on the left-hand side and the chain rule on the right-hand side.

- Left-hand side differentiation:
\[ \frac{d}{dx}(x^4 y^5) = 4x^3 y^5 + x^4 \cdot 5y^4 \frac{dy}{dx} \]

- Right-hand side differentiation:
\[ \frac{d}{dx}((x + y)^{m + 1}) = (m + 1)(x + y)^m \left(1 + \frac{dy}{dx}\right) \]

3. **Set the Derivatives Equal**:
\[ 4x^3 y^5 + 5x^4 y^4 \frac{dy}{dx} = (m + 1)(x + y)^m \left(1 + \frac{dy}{dx}\right) \]

4. **Substitute \(\frac{dy}{dx} = \frac{y}{x}\)**:
Substitute the given derivative into the equation:
\[ 4x^3 y^5 + 5x^4 y^4 \left(\frac{y}{x}\right) = (m + 1)(x + y)^m \left(1 + \frac{y}{x}\right) \]

5. **Simplify the Left Side**:
\[ 4x^3 y^5 + 5x^3 y^5 = 9x^3 y^5 \]

6. **Simplify the Right Side**:
\[ (m + 1)(x + y)^m \left(\frac{x + y}{x}\right) = (m + 1)(x + y)^{m + 1} \cdot \frac{1}{x} \]

7. **Equate Both Sides**:
\[ 9x^3 y^5 = (m + 1)(x + y)^{m + 1} \cdot \frac{1}{x} \]

8. **Substituting \(x^4 y^5\) for \((x + y)^{m + 1}\)**:
From the original equation, we know:
\[ (x + y)^{m + 1} = x^4 y^5 \]
Thus:
\[ 9x^3 y^5 = (m + 1) \cdot \frac{x^4 y^5}{x} \]

9. **Cancel \(y^5\) and Rearrange**:
\[ 9x^3 = (m + 1)x^3 \]
Dividing both sides by \(x^3\) (assuming \(x \neq 0\)):
\[ 9 = m + 1 \]

10. **Solve for \(m\)**:
\[ m = 9 - 1 = 8 \]

### Final Answer:
\[ m = 8 \]