The number of lines that are parallel to 2x + 6y + 7 = 0 and have an intercept of length 10 between the coordinate axes is

Options

A. 0
B. 1
C. 2
D. Infinite

Answer

To solve the problem, we need to find the number of lines that are parallel to the line given by the equation \(2x + 6y + 7 = 0\) and have an intercept of length 10 between the coordinate axes.

### Step-by-Step Solution:

1. **Find the slope of the given line**:
The equation of the line is \(2x + 6y + 7 = 0\). We can rewrite this in slope-intercept form \(y = mx + b\).
\[ 6y = -2x - 7 \implies y = -\frac{1}{3}x - \frac{7}{6} \]
The slope \(m\) of the line is \(-\frac{1}{3}\).

**Hint**: To find the slope from the standard form \(Ax + By + C = 0\), rearrange it to \(y = mx + b\).

2. **Determine the form of the parallel lines**:
Lines parallel to the given line will have the same slope. Therefore, the equation of any line parallel to it can be written as:
\[ y = -\frac{1}{3}x + c \]
where \(c\) is the y-intercept.

**Hint**: Parallel lines share the same slope.

3. **Find the x-intercept and y-intercept**:
The x-intercept occurs when \(y = 0\):
\[ 0 = -\frac{1}{3}x + c \implies x = 3c \]
The y-intercept is simply \(c\).

4. **Calculate the length of the intercept**:
The length of the intercept between the x-axis and y-axis can be calculated using the distance formula:
\[ \text{Length} = \sqrt{(3c - 0)^2 + (0 - c)^2} = \sqrt{(3c)^2 + c^2} = \sqrt{9c^2 + c^2} = \sqrt{10c^2} = |c|\sqrt{10} \]
We are given that the length of the intercept is 10:
\[ |c|\sqrt{10} = 10 \]
Dividing both sides by \(\sqrt{10}\):
\[ |c| = \frac{10}{\sqrt{10}} = \sqrt{10} \]

**Hint**: Use the distance formula to find the length of the intercept.

5. **Determine the values of \(c\)**:
Since \(|c| = \sqrt{10}\), we have two possible values for \(c\):
\[ c = \sqrt{10} \quad \text{or} \quad c = -\sqrt{10} \]

**Hint**: The absolute value gives two solutions, one positive and one negative.

6. **Write the equations of the lines**:
The two lines that are parallel to the original line and have the specified intercepts are:
\[ y = -\frac{1}{3}x + \sqrt{10} \quad \text{and} \quad y = -\frac{1}{3}x - \sqrt{10} \]

**Hint**: Substitute the values of \(c\) back into the line equation to find the specific lines.

7. **Conclusion**:
Since we found two distinct values for \(c\), there are **2 lines** that satisfy the conditions of the problem.

**Final Answer**: The number of lines that are parallel to \(2x + 6y + 7 = 0\) and have an intercept of length 10 between the coordinate axes is **2**.