Consider the circle `S: x^2 + y^2 - 4x-1=0` and the line `L: y = 3x - 1`. If the line L cuts the circle at A & B. (i) Length of the chord AB equal (i) The angle subtended by the chord AB in the minor arc of S is (iii). Acute angle between the line L and the circle S is

Options

A. `(pi)/(4)`
B. `(2pi)/(3)`
C. `(3pi)/(4)`
D. `(5pi)/(6)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Rewrite the equation of the circle in standard form
The given equation of the circle is:
\[ x^2 + y^2 - 4x - 1 = 0 \]

We can rearrange this equation to find the center and radius of the circle. Completing the square for the \(x\) terms:

1. Group the \(x\) terms:
\[ (x^2 - 4x) + y^2 = 1 \]

2. Complete the square for \(x^2 - 4x\):
\[ (x - 2)^2 - 4 + y^2 = 1 \]
\[ (x - 2)^2 + y^2 = 5 \]

From this, we can see that the center of the circle is at \( (2, 0) \) and the radius \( r \) is \( \sqrt{5} \).

### Step 2: Find the points of intersection (A and B) of the line and the circle
The equation of the line is given as:
\[ y = 3x - 1 \]

Substituting this into the circle's equation:
\[ x^2 + (3x - 1)^2 - 4x - 1 = 0 \]

Expanding:
\[ x^2 + (9x^2 - 6x + 1) - 4x - 1 = 0 \]
\[ 10x^2 - 10x = 0 \]

Factoring out \(10x\):
\[ 10x(x - 1) = 0 \]

Thus, \( x = 0 \) or \( x = 1 \).

Now, substituting back to find \(y\):
- For \(x = 0\):
\[ y = 3(0) - 1 = -1 \]
So, point A is \( (0, -1) \).

- For \(x = 1\):
\[ y = 3(1) - 1 = 2 \]
So, point B is \( (1, 2) \).

### Step 3: Calculate the length of chord AB
Using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Where \( A(0, -1) \) and \( B(1, 2) \):
\[ AB = \sqrt{(1 - 0)^2 + (2 - (-1))^2} \]
\[ = \sqrt{1^2 + (2 + 1)^2} \]
\[ = \sqrt{1 + 9} \]
\[ = \sqrt{10} \]

### Step 4: Find the angle subtended by chord AB at the center of the circle
Using the sine rule:
Let \( O \) be the center of the circle \( (2, 0) \). The distances \( OA \) and \( OB \) are both equal to the radius \( \sqrt{5} \).

Using the triangle \( OAB \):
- \( AB = \sqrt{10} \)
- \( OA = OB = \sqrt{5} \)

Using the cosine rule:
\[ AB^2 = OA^2 + OB^2 - 2 \cdot OA \cdot OB \cdot \cos(\theta) \]
\[ 10 = 5 + 5 - 2 \cdot 5 \cdot \cos(\theta) \]
\[ 10 = 10 - 10 \cdot \cos(\theta) \]
\[ 10 \cdot \cos(\theta) = 0 \]
\[ \cos(\theta) = 0 \]
Thus, \( \theta = 90^\circ \).

### Step 5: Find the acute angle between the line L and the radius at point A
The slope of line \( L \) is \( 3 \) and the slope of the radius \( OA \) can be calculated as:
\[ \text{slope of } OA = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 0}{0 - 2} = \frac{-1}{-2} = \frac{1}{2} \]

Using the formula for the angle between two lines:
\[ \tan(\phi) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
Where \( m_1 = 3 \) (slope of line L) and \( m_2 = \frac{1}{2} \):
\[ \tan(\phi) = \left| \frac{3 - \frac{1}{2}}{1 + 3 \cdot \frac{1}{2}} \right| = \left| \frac{\frac{6}{2} - \frac{1}{2}}{1 + \frac{3}{2}} \right| = \left| \frac{\frac{5}{2}}{\frac{5}{2}} \right| = 1 \]

Thus, \( \phi = 45^\circ \).

### Summary of Answers:
1. Length of chord AB = \( \sqrt{10} \)
2. Angle subtended by chord AB at the center = \( 90^\circ \)
3. Acute angle between line L and radius at point A = \( 45^\circ \)