The number of ordered pair(s) (x,y) which satisfy `y=tan^(-1) tan x` and `16(x^2+y^2)-48 pi x +16 pi y +31 pi^2=0` is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to find the number of ordered pairs \((x, y)\) that satisfy the equations:

1. \(y = \tan^{-1}(\tan x)\)
2. \(16(x^2 + y^2) - 48\pi x + 16\pi y + 31\pi^2 = 0\)

### Step 1: Simplify the second equation

We start by dividing the entire equation by 16:

\[ x^2 + y^2 - 3\pi x + \pi y + \frac{31\pi^2}{16} = 0 \]

### Step 2: Identify the form of the equation

This equation can be rearranged to resemble the standard form of a circle:

\[ x^2 + y^2 - 3\pi x + \pi y = -\frac{31\pi^2}{16} \]

### Step 3: Complete the square for \(x\) and \(y\)

To convert this into the standard circle form, we complete the square for both \(x\) and \(y\).

For \(x\):
\[ x^2 - 3\pi x = \left(x - \frac{3\pi}{2}\right)^2 - \frac{9\pi^2}{4} \]

For \(y\):
\[ y^2 + \pi y = \left(y + \frac{\pi}{2}\right)^2 - \frac{\pi^2}{4} \]

### Step 4: Substitute back into the equation

Substituting these back into the equation gives:

\[ \left(x - \frac{3\pi}{2}\right)^2 - \frac{9\pi^2}{4} + \left(y + \frac{\pi}{2}\right)^2 - \frac{\pi^2}{4} = -\frac{31\pi^2}{16} \]

Combining the constants:

\[ \left(x - \frac{3\pi}{2}\right)^2 + \left(y + \frac{\pi}{2}\right)^2 = -\frac{31\pi^2}{16} + \frac{10\pi^2}{4} = -\frac{31\pi^2}{16} + \frac{40\pi^2}{16} = \frac{9\pi^2}{16} \]

### Step 5: Identify the center and radius of the circle

The center of the circle is at \(\left(\frac{3\pi}{2}, -\frac{\pi}{2}\right)\) and the radius \(r\) is:

\[ r = \sqrt{\frac{9\pi^2}{16}} = \frac{3\pi}{4} \]

### Step 6: Analyze the function \(y = \tan^{-1}(\tan x)\)

The function \(y = \tan^{-1}(\tan x)\) is periodic with a period of \(\pi\) and has a range of \(-\frac{\pi}{2} < y < \frac{\pi}{2}\).

### Step 7: Determine intersections

Next, we need to find how many times the circle intersects with the line \(y = \tan^{-1}(\tan x)\) within the bounds of the circle's center and radius.

The center of the circle is at \(\left(\frac{3\pi}{2}, -\frac{\pi}{2}\right)\) and the radius is \(\frac{3\pi}{4}\). The circle extends from:

- \(x = \frac{3\pi}{2} - \frac{3\pi}{4} = \frac{3\pi}{4}\)
- \(x = \frac{3\pi}{2} + \frac{3\pi}{4} = \frac{9\pi}{4}\)

### Step 8: Count the number of intersections

The function \(y = \tan^{-1}(\tan x)\) will intersect the circle at three distinct points within the range of \(x\) values from \(\frac{3\pi}{4}\) to \(\frac{9\pi}{4}\).

### Conclusion

Thus, the number of ordered pairs \((x, y)\) that satisfy both equations is:

\[ \boxed{3} \]