Does `(a)/(x-y)+(b)/(y-z)+(c)/(z-x)=0` represents a pair of planes?

Answer

To determine whether the equation

\[ \frac{a}{x-y} + \frac{b}{y-z} + \frac{c}{z-x} = 0 \]

represents a pair of planes, we need to manipulate it into a form that can be analyzed. The general equation for a pair of planes in three-dimensional space is given by:

\[ Ax^2 + By^2 + Cz^2 + 2Fyz + 2Gzx + 2Hyx = 0 \]

### Step 1: Rewrite the Given Equation

Start by rewriting the given equation in a more manageable form. We can eliminate the denominators by multiplying through by \((x-y)(y-z)(z-x)\):

\[ a(y-z)(z-x) + b(z-x)(x-y) + c(x-y)(y-z) = 0 \]

### Step 2: Expand the Equation

Now, we expand each term:

1. \( a(y-z)(z-x) = a(yz - yx - z^2 + zx) \)
2. \( b(z-x)(x-y) = b(zx - zy - x^2 + xy) \)
3. \( c(x-y)(y-z) = c(xy - xz - y^2 + yz) \)

Combining these expansions gives us a polynomial in \(x\), \(y\), and \(z\).

### Step 3: Collect Like Terms

Rearranging the terms, we can group them by powers of \(x\), \(y\), and \(z\):

\[ Ax^2 + By^2 + Cz^2 + 2Fyz + 2Gzx + 2Hyx = 0 \]

Where \(A\), \(B\), \(C\), \(F\), \(G\), and \(H\) are coefficients that we will determine from our expanded equation.

### Step 4: Identify Coefficients

From the expanded equation, identify the coefficients \(A\), \(B\), \(C\), \(F\), \(G\), and \(H\). This might involve some algebraic manipulation and simplification.

### Step 5: Check the Condition for Pair of Planes

The condition for the equation to represent a pair of planes is:

\[ A + B + C + 2FG - AF^2 - BG^2 - CH^2 = 0 \]

Substituting the identified coefficients into this condition will allow us to check if the equation represents a pair of planes.

### Step 6: Conclusion

If the condition is satisfied (i.e., equals zero), then the equation indeed represents a pair of planes. If not, it does not.