The equation of straight line such that it passes through point `(12,-1)` and the sum of the intercepted made on the axes is equal to 7 is:

Options

A. `2x-y+14=0`
B. `x-2y+14=0`
C. `x-2y=14`
D. `x+6y+6=0`

Answer

To find the equation of the straight line that passes through the point (12, -1) and has the sum of the intercepts on the axes equal to 7, we can follow these steps:

### Step 1: Understand the intercept form of the line
The equation of a straight 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 equation based on the intercepts
From the problem, we know that the sum of the intercepts is:
\[ a + b = 7 \]
This can be rearranged to express \(b\) in terms of \(a\):
\[ b = 7 - a \]

### Step 3: Substitute the point into the intercept form
Since the line passes through the point (12, -1), we substitute \(x = 12\) and \(y = -1\) into the intercept form:
\[ \frac{12}{a} + \frac{-1}{b} = 1 \]
Substituting \(b = 7 - a\) into the equation gives:
\[ \frac{12}{a} - \frac{1}{7 - a} = 1 \]

### Step 4: Clear the fractions
To eliminate the fractions, we can multiply through by \(a(7 - a)\):
\[ 12(7 - a) - a = a(7 - a) \]
Expanding both sides results in:
\[ 84 - 12a - a = 7a - a^2 \]
This simplifies to:
\[ 84 - 13a = 7a - a^2 \]

### Step 5: Rearranging to form a quadratic equation
Rearranging the equation gives:
\[ a^2 - 20a + 84 = 0 \]

### Step 6: Factor the quadratic equation
Now we will factor the quadratic:
\[ (a - 14)(a - 6) = 0 \]
This gives us two possible values for \(a\):
\[ a = 14 \quad \text{or} \quad a = 6 \]

### Step 7: Find corresponding values of \(b\)
Using \(b = 7 - a\):
1. If \(a = 14\):
\[ b = 7 - 14 = -7 \]
2. If \(a = 6\):
\[ b = 7 - 6 = 1 \]

### Step 8: Write the equations of the lines
Now we can write the equations of the lines using the intercept form:
1. For \(a = 14\) and \(b = -7\):
\[ \frac{x}{14} + \frac{y}{-7} = 1 \implies \frac{x}{14} - \frac{y}{7} = 1 \implies x - 2y = 14 \]
2. For \(a = 6\) and \(b = 1\):
\[ \frac{x}{6} + \frac{y}{1} = 1 \implies \frac{x}{6} + y = 1 \implies x + 6y = 6 \]

### Conclusion
The equations of the lines are:
1. \(x - 2y = 14\)
2. \(x + 6y = 6\)