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

Options

A. 1
B. 2
C. 4
D. infinitely many

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 10 between the coordinate axes.

### Step-by-Step Solution:

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

**Hint:** To find the slope from the standard form, rearrange the equation to \(y = mx + b\).

2. **Equation of parallel lines:**
Lines that are parallel will have the same slope. Therefore, any line parallel to the given line can be expressed as:
\[ y = -\frac{1}{3}x + c \]
where \(c\) is a constant.

**Hint:** Remember that parallel lines share the same slope.

3. **Finding the intercepts:**
The intercept form of a line can be expressed as:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
where \(a\) is the x-intercept and \(b\) is the y-intercept. Given that the sum of the intercepts \(a + b = 10\).

4. **Expressing \(b\) in terms of \(a\):**
Since \(a + b = 10\), we can express \(b\) as:
\[ b = 10 - a \]

**Hint:** Use the relationship between intercepts to express one in terms of the other.

5. **Finding the equation of the line using intercepts:**
The equation of the line in intercept form becomes:
\[ \frac{x}{a} + \frac{y}{10 - a} = 1 \]
To find the slope of this line, we can rearrange it:
\[ y = -(10 - a)\frac{x}{a} + (10 - a) \]
The slope of this line is:
\[ m = -\frac{10 - a}{a} \]

6. **Setting the slopes equal:**
Since the lines are parallel, we set the slopes equal:
\[ -\frac{10 - a}{a} = -\frac{1}{3} \]
Cross-multiplying gives:
\[ 3(10 - a) = a \implies 30 - 3a = a \implies 30 = 4a \implies a = \frac{30}{4} = 7.5 \]

**Hint:** When setting slopes equal, ensure to cross-multiply correctly.

7. **Finding \(b\):**
Now substituting \(a = 7.5\) back into \(b = 10 - a\):
\[ b = 10 - 7.5 = 2.5 \]

8. **Identifying the lines:**
The intercepts are \(a = 7.5\) and \(b = 2.5\). This line exists in the first quadrant.
Additionally, we can have a line in the third quadrant with intercepts \(-7.5\) and \(-2.5\).

9. **Conclusion:**
Therefore, there are two lines that are parallel to the given line and have an intercept of 10 between the coordinate axes: one in the first quadrant and one in the third quadrant.

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