The intercepts made by the plane `3x - 2y + 4z =12` on the coordinates axes are:

Options

A. `6,-4,3`
B. `4, -6, 3`
C. `2, -3, 4`
D. `1/4, -1/6, 1/3`

Answer

To find the intercepts made by the plane given by the equation \(3x - 2y + 4z = 12\) on the coordinate axes, we can follow these steps:

### Step 1: Write the equation in intercept form
The general intercept form of a plane is given by:
\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \]
where \(a\), \(b\), and \(c\) are the x-intercept, y-intercept, and z-intercept respectively. We need to manipulate the given equation into this form.

### Step 2: Rearranging the equation
Start with the equation:
\[ 3x - 2y + 4z = 12 \]
We can rearrange it to isolate the terms involving \(x\), \(y\), and \(z\):
\[ 3x = 12 + 2y - 4z \]
Now, divide the entire equation by 12:
\[ \frac{3x}{12} - \frac{2y}{12} + \frac{4z}{12} = 1 \]

### Step 3: Simplifying the equation
Now simplify each term:
\[ \frac{x}{4} - \frac{y}{6} + \frac{z}{3} = 1 \]
Rearranging gives:
\[ \frac{x}{4} + \frac{y}{-6} + \frac{z}{3} = 1 \]

### Step 4: Identifying the intercepts
From the equation \(\frac{x}{4} + \frac{y}{-6} + \frac{z}{3} = 1\), we can identify the intercepts:
- The x-intercept \(a = 4\) (set \(y = 0\) and \(z = 0\))
- The y-intercept \(b = -6\) (set \(x = 0\) and \(z = 0\))
- The z-intercept \(c = 3\) (set \(x = 0\) and \(y = 0\))

### Conclusion
Thus, the intercepts made by the plane on the coordinate axes are:
- x-intercept: \(4\)
- y-intercept: \(-6\)
- z-intercept: \(3\)