Find the equation of the striaght line which passes through the point (3, -2) and cuts off positive intercepts on the x and y-axes which are in the ratio `4 : 3`.

Answer

To find the equation of the straight line that passes through the point (3, -2) and cuts off positive intercepts on the x and y axes in the ratio 4:3, we can follow these steps:

### Step 1: Define the x-intercept and y-intercept
Let the x-intercept be \( a \) and the y-intercept be \( b \). According to the problem, the ratio of the intercepts is given as:
\[ \frac{a}{b} = \frac{4}{3} \]
From this ratio, we can express \( a \) in terms of \( b \):
\[ a = \frac{4}{3}b \]

### Step 2: Write the equation of the line in intercept form
The intercept form of the equation of a line is given by:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]

### Step 3: Substitute the point (3, -2) into the equation
Since the line passes through the point (3, -2), we can substitute \( x = 3 \) and \( y = -2 \) into the intercept form equation:
\[ \frac{3}{a} + \frac{-2}{b} = 1 \]

### Step 4: Clear the fractions
To eliminate the fractions, we can multiply through by \( ab \):
\[ 3b - 2a = ab \]

### Step 5: Substitute \( a \) in terms of \( b \)
Now, substitute \( a = \frac{4}{3}b \) into the equation:
\[ 3b - 2\left(\frac{4}{3}b\right) = \left(\frac{4}{3}b\right)b \]
This simplifies to:
\[ 3b - \frac{8}{3}b = \frac{4}{3}b^2 \]

### Step 6: Combine like terms
To combine the terms on the left side, we can express \( 3b \) as \( \frac{9}{3}b \):
\[ \frac{9}{3}b - \frac{8}{3}b = \frac{4}{3}b^2 \]
This simplifies to:
\[ \frac{1}{3}b = \frac{4}{3}b^2 \]

### Step 7: Multiply through by 3 to eliminate the fraction
Multiplying through by 3 gives:
\[ b = 4b^2 \]

### Step 8: Rearrange the equation
Rearranging gives:
\[ 4b^2 - b = 0 \]
Factoring out \( b \):
\[ b(4b - 1) = 0 \]
This gives us two solutions:
1. \( b = 0 \) (not valid since we need positive intercepts)
2. \( 4b - 1 = 0 \) which leads to \( b = \frac{1}{4} \)

### Step 9: Find \( a \) using \( b \)
Now, substitute \( b = \frac{1}{4} \) back into the equation for \( a \):
\[ a = \frac{4}{3}b = \frac{4}{3} \cdot \frac{1}{4} = \frac{1}{3} \]

### Step 10: Write the final equation of the line
Now that we have \( a \) and \( b \), we can write the equation of the line:
\[ \frac{x}{\frac{1}{3}} + \frac{y}{\frac{1}{4}} = 1 \]
Multiplying through by 12 (the least common multiple of the denominators) gives:
\[ 12 \left(\frac{x}{\frac{1}{3}}\right) + 12 \left(\frac{y}{\frac{1}{4}}\right) = 12 \]
This simplifies to:
\[ 36x + 12y = 12 \]
Dividing through by 12 gives:
\[ 3x + y = 1 \]

### Final Answer
The equation of the straight line is:
\[ 3x + y = 1 \]