A line passes through the point (3, 4) and cuts off intercepts, from the coordinates axes such that their sum is 14. The equation of the line is :

Options

A. 4 x - 3y = 24
B. 4x + 3y = 24
C. 3 x - 4y = 24
D. 3x + 4y = 24

Answer

To find the equation of the line that passes through the point (3, 4) and cuts off intercepts from the coordinate axes such that their sum is 14, we can follow these steps:

### Step 1: Understand the Intercept Form of the Line
The equation of a line in intercept form is given by:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
where \(a\) is the x-intercept and \(b\) is the y-intercept.

### Step 2: Set Up the Relationship Between Intercepts
According to the problem, the sum of the intercepts is 14:
\[ a + b = 14 \]
From this, we can express \(b\) in terms of \(a\):
\[ b = 14 - a \]

### Step 3: Substitute the Point into the Equation
The line passes through the point (3, 4). We can substitute \(x = 3\) and \(y = 4\) into the intercept form of the equation:
\[ \frac{3}{a} + \frac{4}{b} = 1 \]
Now substitute \(b = 14 - a\) into the equation:
\[ \frac{3}{a} + \frac{4}{14 - a} = 1 \]

### Step 4: Solve for \(a\)
To solve this equation, we will first find a common denominator:
\[ \frac{3(14 - a) + 4a}{a(14 - a)} = 1 \]
This simplifies to:
\[ 3(14 - a) + 4a = a(14 - a) \]
Expanding both sides:
\[ 42 - 3a + 4a = 14a - a^2 \]
Combining like terms gives:
\[ 42 + a = 14a - a^2 \]
Rearranging this leads to:
\[ a^2 - 13a + 42 = 0 \]

### Step 5: Factor the Quadratic Equation
Now we will factor the quadratic:
\[ (a - 6)(a - 7) = 0 \]
Thus, \(a = 6\) or \(a = 7\).

### Step 6: Find Corresponding \(b\) Values
Using \(a + b = 14\):
- If \(a = 6\), then \(b = 14 - 6 = 8\).
- If \(a = 7\), then \(b = 14 - 7 = 7\).

### Step 7: Write the Equations of the Lines
For \(a = 6\) and \(b = 8\):
\[ \frac{x}{6} + \frac{y}{8} = 1 \]
Multiplying through by 24 (the LCM of 6 and 8):
\[ 4x + 3y = 24 \]

For \(a = 7\) and \(b = 7\):
\[ \frac{x}{7} + \frac{y}{7} = 1 \]
Multiplying through by 7:
\[ x + y = 7 \]

### Conclusion
The equation of the line that passes through the point (3, 4) and has intercepts whose sum is 14 is:
\[ 4x + 3y = 24 \]