The equation of straight line passing through point (1,2) and having intercept of length 3 between straight line 3x+4y=12 and 3x+4y=24 is

Options

A. 7x + 24y - 55 = 0
B. 24x + 7y - 38 = 0
C. 24x - 7y - 10 = 0
D. 7x - 24y + 41 = 0

Answer

To solve the problem of finding the equation of a straight line passing through the point (1, 2) and having an intercept of length 3 between the lines \(3x + 4y = 12\) and \(3x + 4y = 24\), we can follow these steps:

### Step 1: Identify the equations of the given lines
The two lines are:
1. \(3x + 4y = 12\) (Line 1)
2. \(3x + 4y = 24\) (Line 2)

### Step 2: Determine the distance between the two parallel lines
The distance \(d\) between two parallel lines of the form \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is given by the formula:
\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]
For our lines:
- \(C_1 = 12\)
- \(C_2 = 24\)
- \(A = 3\), \(B = 4\)

Calculating the distance:
\[ d = \frac{|24 - 12|}{\sqrt{3^2 + 4^2}} = \frac{12}{\sqrt{9 + 16}} = \frac{12}{\sqrt{25}} = \frac{12}{5} = 2.4 \]

### Step 3: Find the slope of the required line
Since the required line has an intercept of length 3 between the two given lines, we can set up the relationship:
\[ \text{Length of intercept} = \frac{3}{d} \cdot \text{Distance between the lines} \]
Given that the distance is \(2.4\), we can use the proportionality of the intercepts to find the slope \(m\) of the required line.

### Step 4: Use the point-slope form of the line
The equation of a line in point-slope form is given by:
\[ y - y_1 = m(x - x_1) \]
Substituting the point (1, 2):
\[ y - 2 = m(x - 1) \]
This simplifies to:
\[ y = mx - m + 2 \]

### Step 5: Find the intersection points with the given lines
We will find the intersection of this line with both given lines.

**Intersection with Line 1:**
Substituting \(y = mx - m + 2\) into \(3x + 4y = 12\):
\[ 3x + 4(mx - m + 2) = 12 \]
Expanding:
\[ 3x + 4mx - 4m + 8 = 12 \]
Rearranging gives:
\[ (3 + 4m)x = 4m + 4 \]
Thus:
\[ x_A = \frac{4(m + 1)}{3 + 4m} \]

**Finding \(y_A\):**
Substituting \(x_A\) back into the line equation:
\[ y_A = m\left(\frac{4(m + 1)}{3 + 4m}\right) - m + 2 \]

**Intersection with Line 2:**
Similarly, substituting into \(3x + 4y = 24\):
\[ 3x + 4(mx - m + 2) = 24 \]
This gives:
\[ (3 + 4m)x = 16 + 4m \]
Thus:
\[ x_B = \frac{4(m + 4)}{3 + 4m} \]

**Finding \(y_B\):**
Substituting \(x_B\) back into the line equation:
\[ y_B = m\left(\frac{4(m + 4)}{3 + 4m}\right) - m + 2 \]

### Step 6: Calculate the distance between points A and B
The distance \(AB\) can be calculated using the distance formula:
\[ AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \]
Setting this equal to 3 and solving for \(m\) will yield the slope.

### Step 7: Solve for \(m\) and find the equation of the line
After finding \(m\), substitute it back into the point-slope form to find the equation of the line.

### Final Equation
After performing the calculations and simplifications, we find that the equation of the required line is:
\[ 7x - 24y + 41 = 0 \]