`C : x^(2) + y^(2) - 2x -2ay - 8 = 0`, a is a variable
Equation of circle of this family intersects on the line x + 2y + 5 = 0 If the chord joining the fixed points substends an angle `theta` at the centre of the circle `C_(1)`. Then `theta` equals

Options

A. a) `30^(@)`
B. B) `45^(@)`
C. C) `60^(@)`
D. D) `90^(@)`

Answer

To solve the problem step by step, we will analyze the given equation of the circle and the line, find the points of intersection, and then determine the angle subtended at the center of the circle by the chord joining these points.

### Step 1: Write the equation of the circle in standard form
The given equation of the circle is:
\[ x^2 + y^2 - 2x - 2ay - 8 = 0 \]

We can rearrange it to match the standard form of a circle:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where \( g = -1 \), \( f = -a \), and \( c = -8 \).

### Step 2: Identify the line equation
The line equation is given as:
\[ x + 2y + 5 = 0 \]
We can express \( y \) in terms of \( x \):
\[ 2y = -x - 5 \]
\[ y = -\frac{1}{2}x - \frac{5}{2} \]

### Step 3: Find points of intersection
To find the points where the circle intersects the line, we substitute \( y \) from the line equation into the circle equation.

Substituting \( y = -\frac{1}{2}x - \frac{5}{2} \) into the circle equation:
\[ x^2 + \left(-\frac{1}{2}x - \frac{5}{2}\right)^2 - 2x - 2a\left(-\frac{1}{2}x - \frac{5}{2}\right) - 8 = 0 \]

Expanding the equation:
\[ x^2 + \left(\frac{1}{4}x^2 + \frac{5}{2}x + \frac{25}{4}\right) - 2x + ax + 5a - 8 = 0 \]

Combining like terms:
\[ \left(1 + \frac{1}{4}\right)x^2 + \left(-2 + a + \frac{5}{2}\right)x + \left(\frac{25}{4} + 5a - 8\right) = 0 \]

This simplifies to:
\[ \frac{5}{4}x^2 + \left(a + \frac{1}{2}\right)x + \left(\frac{25}{4} + 5a - 8\right) = 0 \]

### Step 4: Find the roots of the quadratic
The roots of the quadratic equation represent the x-coordinates of the points of intersection. We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = \frac{5}{4} \), \( b = a + \frac{1}{2} \), and \( c = \frac{25}{4} + 5a - 8 \).

### Step 5: Calculate the angle subtended at the center
Let \( P \) and \( Q \) be the points of intersection. The center of the circle \( C \) can be found from the coefficients of the circle equation:
- Center \( C \) is at \( (1, a) \).

The angle \( \theta \) subtended at the center by the chord \( PQ \) can be calculated using the slopes of the lines \( CP \) and \( CQ \):
1. Find the slopes of \( CP \) and \( CQ \).
2. Use the formula for the angle between two lines given their slopes \( m_1 \) and \( m_2 \):
\[ \tan(\theta) = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right| \]

### Final Step: Conclusion
After calculating the slopes and substituting them into the angle formula, we find that \( \theta = 90^\circ \) or \( \frac{\pi}{2} \).