If two parallel chords of a circle, having diameter 4 units, lie on the opposite sides of the centreand subtend angles `cos^-1(1/7) and sec^-1(7)` at the centre respectively, then the distance betweenthese chords, is: (a) `4/sqrt7` (b) `8/sqrt7` (c) `8/7` (d) `16/7`

Options

A. `( 16)/( 7)`
B. ` ( 8)/( sqrt(7))`
C. `(8)/( 7)`
D. `(4)/(sqrt(7))`

Answer

To solve the problem step by step, we will follow the given information and use trigonometric identities.

### Step 1: Determine the radius of the circle
Given that the diameter of the circle is 4 units, we can find the radius \( r \) as follows:
\[ r = \frac{\text{Diameter}}{2} = \frac{4}{2} = 2 \text{ units} \]

**Hint:** Remember that the radius is half of the diameter.

### Step 2: Identify the angles subtended by the chords
The angles subtended by the chords at the center of the circle are given as:
- First chord subtends an angle \( \theta_1 = \cos^{-1}\left(\frac{1}{7}\right) \)
- Second chord subtends an angle \( \theta_2 = \sec^{-1}(7) \)

**Hint:** Make sure to understand the relationship between secant and cosine: \( \sec \theta = \frac{1}{\cos \theta} \).

### Step 3: Calculate \( \cos \theta_1 \) and \( \cos \theta_2 \)
From the definitions:
\[ \cos \theta_1 = \frac{1}{7} \]
For \( \theta_2 \):
\[ \sec \theta_2 = 7 \implies \cos \theta_2 = \frac{1}{7} \]

**Hint:** Both angles give the same cosine value, which helps in simplifying calculations later.

### Step 4: Find the values of \( \cos \frac{\theta_1}{2} \) and \( \cos \frac{\theta_2}{2} \)
Using the half-angle formula:
\[ \cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}} \]
For \( \theta_1 \):
\[ \cos \frac{\theta_1}{2} = \sqrt{\frac{1 + \frac{1}{7}}{2}} = \sqrt{\frac{\frac{8}{7}}{2}} = \sqrt{\frac{8}{14}} = \sqrt{\frac{4}{7}} = \frac{2}{\sqrt{7}} \]

For \( \theta_2 \):
\[ \cos \frac{\theta_2}{2} = \sqrt{\frac{1 + \frac{1}{7}}{2}} = \frac{2}{\sqrt{7}} \]

**Hint:** The half-angle formula is useful for finding the lengths of segments in the triangle.

### Step 5: Calculate the lengths of segments \( OQ \) and \( OP \)
Using the cosine values found:
- For \( OQ \):
\[ OQ = OC \cdot \cos \frac{\theta_1}{2} = 2 \cdot \frac{2}{\sqrt{7}} = \frac{4}{\sqrt{7}} \]
- For \( OP \):
\[ OP = OA \cdot \cos \frac{\theta_2}{2} = 2 \cdot \frac{2}{\sqrt{7}} = \frac{4}{\sqrt{7}} \]

**Hint:** The segments \( OQ \) and \( OP \) are calculated using the radius and the cosine of half the angle.

### Step 6: Calculate the distance between the chords
The distance \( PQ \) between the two chords is given by:
\[ PQ = OQ + OP = \frac{4}{\sqrt{7}} + \frac{4}{\sqrt{7}} = \frac{8}{\sqrt{7}} \]

**Hint:** When finding the distance between two parallel lines, you can simply add the lengths of the segments from the center to each chord.

### Final Answer
The distance between the two chords is:
\[ \boxed{\frac{8}{\sqrt{7}}} \]