Key Concepts and Formulas

• Distance from the center to a chord: If a chord subtends an angle $\theta$ at the center of a circle with radius $R$, the distance $d$ from the center to the chord is given by $d = R\cos\left(\frac{\theta}{2}\right)$.
• Secant and Cosine relationship: $\sec(x) = \frac{1}{\cos(x)}$.
• Sum of distances: If two parallel chords lie on opposite sides of the center, the distance between them is the sum of their distances from the center.

Step-by-Step Solution

Step 1: Define the angles and radius

Let $\theta_1 = \cos^{-1}\left(\frac{1}{7}\right)$ and $\theta_2 = \sec^{-1}(7)$. The radius of the circle is $R = \frac{4}{2} = 2$.

Step 2: Simplify $\theta_2$

Since $\theta_2 = \sec^{-1}(7)$, we have $\sec(\theta_2) = 7$. Therefore, $\cos(\theta_2) = \frac{1}{\sec(\theta_2)} = \frac{1}{7}$. Thus, $\theta_2 = \cos^{-1}\left(\frac{1}{7}\right)$.

Step 3: Notice the equality of the angles

We observe that $\theta_1 = \cos^{-1}\left(\frac{1}{7}\right) = \theta_2$. Let's denote this common angle by $\theta$, so $\theta = \cos^{-1}\left(\frac{1}{7}\right)$.

Step 4: Calculate the distance of each chord from the center

The distance of the first chord from the center is $d_1 = R \cos\left(\frac{\theta_1}{2}\right) = 2\cos\left(\frac{\theta}{2}\right)$. The distance of the second chord from the center is $d_2 = R \cos\left(\frac{\theta_2}{2}\right) = 2\cos\left(\frac{\theta}{2}\right)$.

Step 5: Calculate $\cos(\theta/2)$

We know that $\cos(\theta) = \frac{1}{7}$. We use the identity $\cos(\theta) = 2\cos^2\left(\frac{\theta}{2}\right) - 1$ to find $\cos\left(\frac{\theta}{2}\right)$. $\frac{1}{7} = 2\cos^2\left(\frac{\theta}{2}\right) - 1$ $2\cos^2\left(\frac{\theta}{2}\right) = \frac{1}{7} + 1 = \frac{8}{7}$ $\cos^2\left(\frac{\theta}{2}\right) = \frac{4}{7}$ $\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{4}{7}} = \frac{2}{\sqrt{7}}$ (Since $0 < \frac{\theta}{2} < \frac{\pi}{2}$, $\cos\left(\frac{\theta}{2}\right)$ is positive).

Step 6: Calculate the distances $d_1$ and $d_2$

$d_1 = 2\cos\left(\frac{\theta}{2}\right) = 2\left(\frac{2}{\sqrt{7}}\right) = \frac{4}{\sqrt{7}}$. $d_2 = 2\cos\left(\frac{\theta}{2}\right) = 2\left(\frac{2}{\sqrt{7}}\right) = \frac{4}{\sqrt{7}}$.

Step 7: Calculate the distance between the chords

Since the chords are on opposite sides of the center, the distance between them is $d = d_1 + d_2 = \frac{4}{\sqrt{7}} + \frac{4}{\sqrt{7}} = \frac{8}{\sqrt{7}}$.

Common Mistakes & Tips

• Remember to halve the angle when calculating the distance from the center to the chord using the formula $d = R\cos\left(\frac{\theta}{2}\right)$.
• Be careful when dealing with inverse trigonometric functions. Convert them to cosine or sine to work with them more easily.
• Don't forget to add the distances from the center when the chords are on opposite sides.

Summary

We found the distance of each chord from the center of the circle using the formula $d = R\cos\left(\frac{\theta}{2}\right)$, where $\theta$ is the angle subtended by the chord at the center. After finding that the angles subtended by both chords were equal, we calculated the distance of each chord from the center and added them to find the total distance between the parallel chords, which is $\frac{8}{\sqrt{7}}$.

Final Answer The final answer is $\boxed{\frac{8}{\sqrt{7}}}$, which corresponds to option (B).