Find the equation of a plane passes through the point `(4,4,1)` and the ratio of intercepts cuts on axes from this plane is `2 : 1 : 1`.

Answer

To find the equation of the plane that passes through the point \( (4, 4, 1) \) and has intercepts on the axes in the ratio \( 2:1:1 \), we can follow these steps:

### Step 1: Understand the intercept form of the equation of a plane
The equation of a plane in intercept form is given by:
\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \]
where \( a \), \( b \), and \( c \) are the x, y, and z-intercepts respectively.

### Step 2: Set the intercepts based on the given ratio
Given that the ratio of the intercepts is \( 2:1:1 \), we can express the intercepts as:
- \( a = 2\lambda \)
- \( b = \lambda \)
- \( c = \lambda \)

### Step 3: Substitute the intercepts into the intercept form
Substituting the values of \( a \), \( b \), and \( c \) into the intercept form, we have:
\[ \frac{x}{2\lambda} + \frac{y}{\lambda} + \frac{z}{\lambda} = 1 \]

### Step 4: Clear the denominators
To eliminate the denominators, multiply through by \( 2\lambda \):
\[ x + 2y + 2z = 2\lambda \]

### Step 5: Use the point through which the plane passes
Since the plane passes through the point \( (4, 4, 1) \), we substitute \( x = 4 \), \( y = 4 \), and \( z = 1 \) into the equation:
\[ 4 + 2(4) + 2(1) = 2\lambda \]

### Step 6: Simplify the equation
Calculating the left-hand side:
\[ 4 + 8 + 2 = 2\lambda \implies 14 = 2\lambda \]

### Step 7: Solve for \( \lambda \)
Dividing both sides by 2 gives:
\[ \lambda = 7 \]

### Step 8: Substitute \( \lambda \) back to find intercepts
Now substituting \( \lambda \) back into the expressions for \( a \), \( b \), and \( c \):
- \( a = 2\lambda = 2(7) = 14 \)
- \( b = \lambda = 7 \)
- \( c = \lambda = 7 \)

### Step 9: Write the final equation of the plane
Substituting \( a \), \( b \), and \( c \) back into the intercept form:
\[ \frac{x}{14} + \frac{y}{7} + \frac{z}{7} = 1 \]

### Step 10: Clear the denominators again
Multiplying through by 14 to eliminate the denominators gives:
\[ x + 2y + 2z = 14 \]

Thus, the equation of the plane is:
\[ \boxed{x + 2y + 2z = 14} \]