If the tangent at P on the curve `x^my^n =d^(m+n)` meets the co-ordinates axes at A and B, then AP : PB=

Options

A. `m:n`
B. `n:m`
C. `-m:n`
D. `-n:m`

Answer

To solve the problem of finding the ratio \( AP : PB \) where the tangent at point \( P \) on the curve \( x^m y^n = d^{m+n} \) meets the coordinate axes at points \( A \) and \( B \), we can follow these steps:

### Step 1: Understand the Curve
The given curve is \( x^m y^n = d^{m+n} \). We can rewrite this in logarithmic form to make differentiation easier.

### Step 2: Take Logarithm
Taking the logarithm of both sides, we have:
\[ \log(x^m y^n) = \log(d^{m+n}) \]
Using the property of logarithms, this simplifies to:
\[ m \log x + n \log y = (m+n) \log d \]

### Step 3: Differentiate Implicitly
Let \( P \) be the point \( (x_1, y_1) \) on the curve. We differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(m \log x + n \log y) = 0 \]
This gives us:
\[ \frac{m}{x} + n \frac{1}{y} \frac{dy}{dx} = 0 \]
Rearranging this, we find:
\[ \frac{dy}{dx} = -\frac{m y}{n x} \]

### Step 4: Find the Slope at Point \( P \)
At point \( P(x_1, y_1) \), the slope \( m \) of the tangent line is:
\[ m = -\frac{m y_1}{n x_1} \]

### Step 5: Write the Equation of the Tangent Line
The equation of the tangent line at point \( P \) can be written as:
\[ y - y_1 = m (x - x_1) \]
Substituting for \( m \):
\[ y - y_1 = -\frac{m y_1}{n x_1} (x - x_1) \]

### Step 6: Find Points \( A \) and \( B \)
To find the x-intercept (point \( A \)), set \( y = 0 \):
\[ 0 - y_1 = -\frac{m y_1}{n x_1} (x - x_1) \]
Solving for \( x \):
\[ x = x_1 + \frac{n x_1}{m} \]
Thus, the coordinates of point \( A \) are:
\[ A \left( x_1 + \frac{n x_1}{m}, 0 \right) \]

To find the y-intercept (point \( B \)), set \( x = 0 \):
\[ y - y_1 = -\frac{m y_1}{n x_1} (0 - x_1) \]
Solving for \( y \):
\[ y = y_1 + \frac{m y_1}{n} \]
Thus, the coordinates of point \( B \) are:
\[ B \left( 0, y_1 + \frac{m y_1}{n} \right) \]

### Step 7: Calculate Distances \( AP \) and \( PB \)
The distance \( AP \) can be calculated as:
\[ AP = \left| x_1 + \frac{n x_1}{m} - x_1 \right| = \frac{n x_1}{m} \]

The distance \( PB \) can be calculated as:
\[ PB = \left| y_1 + \frac{m y_1}{n} - y_1 \right| = \frac{m y_1}{n} \]

### Step 8: Find the Ratio \( AP : PB \)
Now, we can find the ratio:
\[ AP : PB = \frac{\frac{n x_1}{m}}{\frac{m y_1}{n}} = \frac{n^2 x_1}{m^2 y_1} \]
However, since we are looking for the ratio of the segments \( AP \) and \( PB \) directly, we can simplify to:
\[ AP : PB = n : m \]

### Final Answer
Thus, the ratio \( AP : PB \) is:
\[ \boxed{n : m} \]