Find the equation of a plane passing through the points A(a,0,0), B(0,b,0) and C(0,0,c)`.

Explanation

To find the equation of the plane passing through the points A(a, 0, 0), B(0, b, 0), and C(0, 0, c), we can use the intercept form of the equation of a plane. Here’s how to derive it step by step:

### Step 1: Identify the intercepts
The points A, B, and C represent the intercepts on the x-axis, y-axis, and z-axis respectively:
- Point A(a, 0, 0) is the x-intercept.
- Point B(0, b, 0) is the y-intercept.
- Point C(0, 0, c) is the z-intercept.

### Step 2: Write the general 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 3: Substitute the intercepts into the equation
Substituting the intercepts from points A, B, and C into the equation, we get:
\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \]

### Step 4: Rearranging the equation
To express the equation in a more standard form, we can multiply through by \(abc\) (the product of the intercepts) to eliminate the denominators:
\[ bcx + acy + abz = abc \]

### Conclusion
Thus, the equation of the plane passing through the points A(a, 0, 0), B(0, b, 0), and C(0, 0, c) is:
\[ bcx + acy + abz = abc \]

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