Find the equations of the line which passes through the point `(4,4)` and the sum of its intercepts on the axes is`16` .

Answer

To find the equation of the line that passes through the point (4, 4) and whose intercepts on the axes sum up to 16, we can follow these steps:

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

### Step 2: Set up the equation using the given point
Since the line passes through the point (4, 4), we can substitute \(x = 4\) and \(y = 4\) into the intercept form equation:
\[ \frac{4}{a} + \frac{4}{b} = 1 \]

### Step 3: Simplify the equation
Factoring out the common term:
\[ 4\left(\frac{1}{a} + \frac{1}{b}\right) = 1 \]
Dividing both sides by 4:
\[ \frac{1}{a} + \frac{1}{b} = \frac{1}{4} \]

### Step 4: Use the condition on intercepts
We know that the sum of the intercepts is given as:
\[ a + b = 16 \]

### Step 5: Express \(b\) in terms of \(a\)
From the equation \(a + b = 16\), we can express \(b\) as:
\[ b = 16 - a \]

### Step 6: Substitute \(b\) into the intercept equation
Substituting \(b\) into the intercept equation:
\[ \frac{1}{a} + \frac{1}{16 - a} = \frac{1}{4} \]

### Step 7: Clear the fractions
To eliminate the fractions, we can multiply through by \(4a(16 - a)\):
\[ 4(16 - a) + 4a = a(16 - a) \]
This simplifies to:
\[ 64 - 4a + 4a = 16a - a^2 \]
Thus, we have:
\[ 64 = 16a - a^2 \]

### Step 8: Rearrange the equation
Rearranging gives us a standard quadratic equation:
\[ a^2 - 16a + 64 = 0 \]

### Step 9: Factor the quadratic equation
This can be factored as:
\[ (a - 8)^2 = 0 \]
Thus, we find:
\[ a = 8 \]

### Step 10: Find \(b\)
Using \(a + b = 16\):
\[ 8 + b = 16 \implies b = 8 \]

### Step 11: Write the equation of the line
Now substituting \(a\) and \(b\) back into the intercept form:
\[ \frac{x}{8} + \frac{y}{8} = 1 \]
This simplifies to:
\[ x + y = 8 \]

### Final Answer
The equation of the line is:
\[ \boxed{x + y = 8} \]