The set of values of a for which the point `(2a, a + 1)` is an interior point of the larger segment of the circle `x^2 + y^2 - 2x - 2y - 8 = 0` made by the chord `x - y +1 = 0`, is

Options

A. (-1,9/5)
B. (0, 9/5)
C. `(0, oo)`
D. none of these

Explanation

To solve the problem, we need to determine the set of values of \( a \) for which the point \( (2a, a + 1) \) is an interior point of the larger segment of the circle defined by the equation \( x^2 + y^2 - 2x - 2y - 8 = 0 \) and is also on the same side of the chord defined by \( x - y + 1 = 0 \).

### Step 1: Rewrite the Circle Equation
First, we rewrite the equation of the circle in standard form. The given equation is:
\[ x^2 + y^2 - 2x - 2y - 8 = 0 \]
We can complete the square for both \( x \) and \( y \):
\[ (x^2 - 2x) + (y^2 - 2y) = 8 \]
Completing the square:
\[ (x - 1)^2 - 1 + (y - 1)^2 - 1 = 8 \]
\[ (x - 1)^2 + (y - 1)^2 = 10 \]
This represents a circle centered at \( (1, 1) \) with a radius of \( \sqrt{10} \).

### Step 2: Check if the Point is Inside the Circle
Next, we need to check if the point \( (2a, a + 1) \) is inside the circle. For the point to be inside, we must satisfy:
\[ (2a - 1)^2 + (a + 1 - 1)^2 < 10 \]
This simplifies to:
\[ (2a - 1)^2 + a^2 < 10 \]
Expanding this:
\[ (4a^2 - 4a + 1) + a^2 < 10 \]
\[ 5a^2 - 4a + 1 < 10 \]
\[ 5a^2 - 4a - 9 < 0 \]

### Step 3: Solve the Quadratic Inequality
Now we solve the quadratic inequality \( 5a^2 - 4a - 9 < 0 \). First, we find the roots using the quadratic formula:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 5 \cdot (-9)}}{2 \cdot 5} \]
Calculating the discriminant:
\[ = \frac{4 \pm \sqrt{16 + 180}}{10} = \frac{4 \pm \sqrt{196}}{10} = \frac{4 \pm 14}{10} \]
This gives us:
\[ a_1 = \frac{18}{10} = \frac{9}{5}, \quad a_2 = \frac{-10}{10} = -1 \]
Thus, the roots are \( a = -1 \) and \( a = \frac{9}{5} \).

### Step 4: Determine the Interval
The quadratic \( 5a^2 - 4a - 9 \) opens upwards (since the coefficient of \( a^2 \) is positive), so the inequality \( 5a^2 - 4a - 9 < 0 \) holds between the roots:
\[ -1 < a < \frac{9}{5} \]

### Step 5: Check the Position Relative to the Chord
Next, we need to check that the point \( (2a, a + 1) \) is on the same side of the chord \( x - y + 1 = 0 \). We substitute the point into the chord equation:
\[ 2a - (a + 1) + 1 > 0 \]
This simplifies to:
\[ 2a - a - 1 + 1 > 0 \implies a > 0 \]

### Final Step: Combine the Conditions
Now we combine the conditions:
1. From the circle: \( -1 < a < \frac{9}{5} \)
2. From the chord: \( a > 0 \)

Thus, the set of values for \( a \) is:
\[ 0 < a < \frac{9}{5} \]

### Final Answer
The set of values of \( a \) for which the point \( (2a, a + 1) \) is an interior point of the larger segment of the circle is:
\[ (0, \frac{9}{5}) \]