Tangents are drawn from the point (a, a) to the circle `x^(2)+y^(2)-2x-2y-6=0` If the angle between the tangents lies in the range `((pi)/(3), pi)`, then the exhaustive range of values of a is:

Options

A. `(1, oo)`
B. `(-5, -3) cup (3, 5)`
C. `(-oo, 2sqrt(2)) cup (2 sqrt(2), oo)`
D. `(-3, -1) cup (3, 5)`

Answer

To solve the problem, we need to find the exhaustive range of values of \( a \) such that the angle between the tangents drawn from the point \( (a, a) \) to the circle defined by the equation \( x^2 + y^2 - 2x - 2y - 6 = 0 \) lies in the range \( \left(\frac{\pi}{3}, \pi\right) \).

### Step 1: Rewrite the Circle Equation
First, we rewrite the circle equation in standard form. The given equation is:
\[ x^2 + y^2 - 2x - 2y - 6 = 0 \]
We can complete the square for \( x \) and \( y \):
\[ (x^2 - 2x) + (y^2 - 2y) = 6 \]
Completing the square:
\[ (x - 1)^2 - 1 + (y - 1)^2 - 1 = 6 \]
This simplifies to:
\[ (x - 1)^2 + (y - 1)^2 = 8 \]
Thus, the center of the circle is \( (1, 1) \) and the radius \( r \) is \( \sqrt{8} = 2\sqrt{2} \).

### Step 2: Find the Distance from Point to Center
Next, we find the distance \( d \) from the point \( (a, a) \) to the center of the circle \( (1, 1) \):
\[ d = \sqrt{(a - 1)^2 + (a - 1)^2} = \sqrt{2(a - 1)^2} = \sqrt{2}|a - 1| \]

### Step 3: Use the Tangent Angle Formula
The angle \( \theta \) between the tangents from a point outside the circle is given by:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{r}{d} \]
Where \( r = 2\sqrt{2} \) and \( d = \sqrt{2}|a - 1| \). Thus,
\[ \tan\left(\frac{\theta}{2}\right) = \frac{2\sqrt{2}}{\sqrt{2}|a - 1|} = \frac{2}{|a - 1|} \]

### Step 4: Find the Range of \( a \)
We know that the angle \( \theta \) lies in the range \( \left(\frac{\pi}{3}, \pi\right) \). This means:
\[ \frac{\pi}{3} < \theta < \pi \]
This implies:
\[ 0 < \frac{\theta}{2} < \frac{\pi}{2} \]
Thus,
\[ \tan\left(\frac{\pi}{3}\right) < \tan\left(\frac{\theta}{2}\right) < \tan(\pi) \]
Since \( \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \) and \( \tan(\pi) = 0 \), we have:
\[ \sqrt{3} < \frac{2}{|a - 1|} < \infty \]
From \( \sqrt{3} < \frac{2}{|a - 1|} \), we get:
\[ |a - 1| < \frac{2}{\sqrt{3}} \]
This leads to two inequalities:
\[ - \frac{2}{\sqrt{3}} < a - 1 < \frac{2}{\sqrt{3}} \]
Thus, adding 1 to all parts:
\[ 1 - \frac{2}{\sqrt{3}} < a < 1 + \frac{2}{\sqrt{3}} \]

### Step 5: Calculate the Values
Calculating the numerical values:
\[ 1 - \frac{2}{\sqrt{3}} \approx 1 - 1.1547 \approx -0.1547 \]
\[ 1 + \frac{2}{\sqrt{3}} \approx 1 + 1.1547 \approx 2.1547 \]

### Final Answer
Thus, the exhaustive range of values of \( a \) is:
\[ \left(1 - \frac{2}{\sqrt{3}}, 1 + \frac{2}{\sqrt{3}}\right) \approx (-0.1547, 2.1547) \]