Let `alpha` be a variable parameter, then the length of the chord of the curve: `(x-sin^-1 alpha)(x-cos^-1 alpha)+(y-sin^-1 alpha)(y+cos^-1 alpha)=0` along the line `x=pi/4` can not be equal to

Options

A. `(pi)/(3)`
B. `(pi)/(6)`
C. `(pi)/(4)`
D. `(pi)/(2)`

Answer

To solve the problem, we need to find the length of the chord of the given curve along the line \( x = \frac{\pi}{4} \) and determine which length cannot be equal to a certain value.

### Step-by-step Solution:

1. **Substituting \( x = \frac{\pi}{4} \)**:
We start with the given equation of the curve:
\[ (x - \sin^{-1} \alpha)(x - \cos^{-1} \alpha) + (y - \sin^{-1} \alpha)(y + \cos^{-1} \alpha) = 0 \]
We substitute \( x = \frac{\pi}{4} \):
\[ \left(\frac{\pi}{4} - \sin^{-1} \alpha\right)\left(\frac{\pi}{4} - \cos^{-1} \alpha\right) + (y - \sin^{-1} \alpha)(y + \cos^{-1} \alpha) = 0 \]

2. **Simplifying the Equation**:
Let \( A = \frac{\pi}{4} - \sin^{-1} \alpha \) and \( B = \frac{\pi}{4} - \cos^{-1} \alpha \). The equation becomes:
\[ AB + (y - \sin^{-1} \alpha)(y + \cos^{-1} \alpha) = 0 \]
Expanding the second term:
\[ AB + y^2 + y \cos^{-1} \alpha - y \sin^{-1} \alpha - \sin^{-1} \alpha \cos^{-1} \alpha = 0 \]

3. **Rearranging the Equation**:
Rearranging gives us a quadratic in \( y \):
\[ y^2 + ( \cos^{-1} \alpha - \sin^{-1} \alpha ) y + (AB - \sin^{-1} \alpha \cos^{-1} \alpha) = 0 \]

4. **Finding the Roots**:
Let \( y_1 \) and \( y_2 \) be the roots of this quadratic equation. By Vieta's formulas:
- \( y_1 + y_2 = -b/a = -(\cos^{-1} \alpha - \sin^{-1} \alpha) \)
- \( y_1 y_2 = c/a = AB - \sin^{-1} \alpha \cos^{-1} \alpha \)

5. **Calculating the Length of the Chord**:
The length of the chord is given by:
\[ |y_2 - y_1| = \sqrt{(y_1 + y_2)^2 - 4y_1 y_2} \]
Substituting the values from Vieta's:
\[ |y_2 - y_1| = \sqrt{(-(\cos^{-1} \alpha - \sin^{-1} \alpha))^2 - 4(AB - \sin^{-1} \alpha \cos^{-1} \alpha)} \]

6. **Finding Minimum Value**:
We need to analyze the expression to find the minimum value of \( |y_2 - y_1| \). We know:
\[ \sin^{-1} \alpha + \cos^{-1} \alpha = \frac{\pi}{2} \]
Therefore, we can express \( \cos^{-1} \alpha \) in terms of \( \sin^{-1} \alpha \):
\[ \cos^{-1} \alpha = \frac{\pi}{2} - \sin^{-1} \alpha \]

7. **Final Calculation**:
After substituting and simplifying, we find that the minimum value of \( |y_2 - y_1| \) is \( \frac{\pi}{2} \).

### Conclusion:
The length of the chord cannot be equal to values less than \( \frac{\pi}{2} \). Therefore, the values that cannot be equal to the length of the chord include \( \frac{\pi}{3} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{6} \).