The real value of a for which the valye of m satisfying the equation `(a^(2)-1)m^(2)-(2a-3)m +a =0` gives the slope of a line parallel to the y-axis is

Options

A. `(3)/(2)`
B. 0
C. 1
D. `+-1`

Explanation

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

\[ (a^2 - 1)m^2 - (2a - 3)m + a = 0 \]

gives the slope of a line parallel to the y-axis, we need to analyze the conditions under which the slope is infinite.

### Step 1: Understand the condition for a slope parallel to the y-axis
A line that is parallel to the y-axis has an undefined slope, which can be interpreted as having a vertical line. In terms of the quadratic equation, this means that one of the roots of the equation must be infinite.

### Step 2: Identify when the roots are infinite
For a quadratic equation of the form \( Ax^2 + Bx + C = 0 \), the roots can be infinite if the coefficient \( A = 0 \) (since the equation would not be quadratic anymore) or if the discriminant \( B^2 - 4AC = 0 \) (which leads to a double root).

### Step 3: Set the coefficient of \( m^2 \) to zero
To find when the slope is infinite, we first set the coefficient of \( m^2 \) to zero:

\[ a^2 - 1 = 0 \]

This gives us:

\[ a^2 = 1 \implies a = 1 \text{ or } a = -1 \]

### Step 4: Set the discriminant to zero
Next, we need to check the discriminant condition:

\[ B^2 - 4AC = 0 \]

Here, \( A = a^2 - 1 \), \( B = -(2a - 3) \), and \( C = a \). Thus, we compute:

\[ (-(2a - 3))^2 - 4(a^2 - 1)(a) = 0 \]

Expanding this:

\[ (2a - 3)^2 - 4(a^2 - 1)a = 0 \]

Calculating \( (2a - 3)^2 \):

\[ 4a^2 - 12a + 9 \]

Calculating \( 4(a^2 - 1)a \):

\[ 4a^3 - 4a \]

Setting the equation:

\[ 4a^2 - 12a + 9 - (4a^3 - 4a) = 0 \]

Rearranging gives:

\[ 4a^3 - 4a^2 - 8a + 9 = 0 \]

### Step 5: Solve the cubic equation
To find the values of \( a \), we can use numerical methods or factorization. For simplicity, we can test for rational roots using the Rational Root Theorem. Testing \( a = 1 \):

\[ 4(1)^3 - 4(1)^2 - 8(1) + 9 = 4 - 4 - 8 + 9 = 1 \quad \text{(not a root)} \]

Testing \( a = 2 \):

\[ 4(2)^3 - 4(2)^2 - 8(2) + 9 = 32 - 16 - 16 + 9 = 9 \quad \text{(not a root)} \]

Testing \( a = 3 \):

\[ 4(3)^3 - 4(3)^2 - 8(3) + 9 = 108 - 36 - 24 + 9 = 57 \quad \text{(not a root)} \]

Continuing this process, we find:

Testing \( a = \frac{3}{2} \):

\[ 4\left(\frac{3}{2}\right)^3 - 4\left(\frac{3}{2}\right)^2 - 8\left(\frac{3}{2}\right) + 9 = 4 \cdot \frac{27}{8} - 4 \cdot \frac{9}{4} - 12 + 9 = \frac{27}{2} - 9 - 12 + 9 = 0 \]

Thus, \( a = \frac{3}{2} \) is also a solution.

### Conclusion
The values of \( a \) for which the slope of the line is parallel to the y-axis are:

\[ a = 1, \quad a = -1, \quad a = \frac{3}{2} \]