For what values of m will the equation `x^(2)-2mx+7m-12=0` have (i) equal roots, (ii) reciprocal roots ?

Answer

To solve the problem, we need to find the values of \( m \) for which the quadratic equation \( x^2 - 2mx + (7m - 12) = 0 \) has (i) equal roots and (ii) reciprocal roots.

### Step-by-Step Solution

#### Part (i): Equal Roots

1. **Identify the coefficients**:
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \).
- Here, \( a = 1 \), \( b = -2m \), and \( c = 7m - 12 \).

2. **Use the condition for equal roots**:
For a quadratic equation to have equal roots, the discriminant must be zero:
\[ D = b^2 - 4ac = 0 \]

3. **Calculate the discriminant**:
Substitute the values of \( a \), \( b \), and \( c \) into the discriminant formula:
\[ D = (-2m)^2 - 4(1)(7m - 12) \]
Simplifying this gives:
\[ D = 4m^2 - 4(7m - 12) \]
\[ D = 4m^2 - 28m + 48 \]

4. **Set the discriminant to zero**:
\[ 4m^2 - 28m + 48 = 0 \]

5. **Divide the entire equation by 4**:
\[ m^2 - 7m + 12 = 0 \]

6. **Factor the quadratic equation**:
We can factor it as:
\[ (m - 3)(m - 4) = 0 \]

7. **Find the values of \( m \)**:
Setting each factor to zero gives:
\[ m - 3 = 0 \quad \Rightarrow \quad m = 3 \]
\[ m - 4 = 0 \quad \Rightarrow \quad m = 4 \]

Thus, the values of \( m \) for which the equation has equal roots are \( m = 3 \) and \( m = 4 \).

#### Part (ii): Reciprocal Roots

1. **Use the condition for reciprocal roots**:
If the roots are \( \alpha \) and \( \frac{1}{\alpha} \), then the product of the roots is given by:
\[ \alpha \cdot \frac{1}{\alpha} = 1 \]

2. **Relate the product of the roots to the coefficients**:
The product of the roots can also be expressed as:
\[ \frac{c}{a} = \frac{7m - 12}{1} = 7m - 12 \]
Setting this equal to 1 gives:
\[ 7m - 12 = 1 \]

3. **Solve for \( m \)**:
Rearranging the equation gives:
\[ 7m = 13 \]
\[ m = \frac{13}{7} \]

Thus, the value of \( m \) for which the equation has reciprocal roots is \( m = \frac{13}{7} \).

### Final Answers
- For equal roots: \( m = 3 \) and \( m = 4 \)
- For reciprocal roots: \( m = \frac{13}{7} \)