The equation of the line through `(4,1)`, whose x-intercept is double its `y-`intercepts on the axes is

Options

A. `x+2y=6`
B. `2x+y=6`
C. `x+2y+6=0`
D. none of these

Answer

To find the equation of the line that passes through the point (4, 1) and has an x-intercept that is double its y-intercept, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Intercepts:**
Let the x-intercept be \( a \) and the y-intercept be \( b \). According to the problem, we have:
\[ a = 2b \]

2. **Use the Intercept Form of the Line:**
The equation of a line in intercept form is given by:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]

3. **Substituting the x-intercept:**
Substituting \( a = 2b \) into the intercept form gives:
\[ \frac{x}{2b} + \frac{y}{b} = 1 \]

4. **Clear the Denominator:**
To eliminate the denominators, multiply the entire equation by \( 2b \):
\[ x + 2y = 2b \]

5. **Substituting the Point (4, 1):**
Since the line passes through the point (4, 1), we can substitute \( x = 4 \) and \( y = 1 \) into the equation:
\[ 4 + 2(1) = 2b \]
Simplifying this gives:
\[ 4 + 2 = 2b \implies 6 = 2b \]

6. **Solve for \( b \):**
Dividing both sides by 2 gives:
\[ b = 3 \]

7. **Find \( a \):**
Now, substituting \( b \) back into the equation for \( a \):
\[ a = 2b = 2(3) = 6 \]

8. **Write the Final Equation:**
Now substituting \( a \) and \( b \) back into the intercept form:
\[ \frac{x}{6} + \frac{y}{3} = 1 \]

9. **Multiply through by 6 to eliminate the fractions:**
\[ x + 2y = 6 \]

### Final Answer:
The equation of the line is:
\[ x + 2y = 6 \]