For what value of m will the equation `(x^2-bx)/(ax-c)=(m-1)/(m+1)` have roots equal in magnitude but opposite in sign?

Options

A. `(a-b)/(a+b)`
B. `(b-a)/(a+b)`
C. `(a+b)/(a-b)`
D. `(b+a)/(b-a)`

Answer

To find the value of \( m \) for which the equation

\[ \frac{x^2 - bx}{ax - c} = \frac{m - 1}{m + 1} \]

has roots that are equal in magnitude but opposite in sign, we can follow these steps:

### Step 1: Rearranging the Equation

Start with the given equation:

\[ \frac{x^2 - bx}{ax - c} = \frac{m - 1}{m + 1} \]

Cross-multiply to eliminate the fraction:

\[ (x^2 - bx)(m + 1) = (ax - c)(m - 1) \]

### Step 2: Expanding Both Sides

Expanding both sides gives:

\[ (m + 1)x^2 - b(m + 1)x = (m - 1)ax - (m - 1)c \]

### Step 3: Rearranging to Form a Quadratic Equation

Rearranging the equation to one side results in:

\[ (m + 1)x^2 - (b(m + 1) + (m - 1)a)x + (m - 1)c = 0 \]

### Step 4: Identifying the Coefficients

This is a standard quadratic equation of the form \( Ax^2 + Bx + C = 0 \), where:

- \( A = m + 1 \)
- \( B = -(b(m + 1) + (m - 1)a) \)
- \( C = (m - 1)c \)

### Step 5: Condition for Roots to be Equal in Magnitude but Opposite in Sign

For the roots to be equal in magnitude but opposite in sign, the sum of the roots must be zero. The sum of the roots is given by:

\[ -\frac{B}{A} = 0 \]

This implies that \( B = 0 \).

### Step 6: Setting B to Zero

Set the coefficient \( B \) to zero:

\[ b(m + 1) + (m - 1)a = 0 \]

### Step 7: Solving for \( m \)

Rearranging gives:

\[ b(m + 1) = -(m - 1)a \]

Expanding and rearranging leads to:

\[ bm + b = -am + a \]

Combining terms gives:

\[ bm + am = a - b \]

Factoring out \( m \):

\[ m(a + b) = a - b \]

Finally, solving for \( m \):

\[ m = \frac{a - b}{a + b} \]

### Conclusion

Thus, the value of \( m \) for which the equation has roots equal in magnitude but opposite in sign is:

\[ m = \frac{a - b}{a + b} \]

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