If two parallel chords of a circle, having diameter 4 units, lie on the opposite sides of the center and subtend angles $$\cos^{-1}\left(\frac{1}{7}\right)$$ and $$\sec^{-1}(7)$$ at the center respectively, then the distance between these chords is:

We have a circle whose diameter is given to be 4 units. The radius is therefore

$$R=\frac{\text{diameter}}{2}= \frac{4}{2}=2.$$

The problem speaks about two chords which are parallel and lie on the opposite sides of the centre O. Let the smaller (acute) central angle made by one chord be

$$\theta_1=\cos^{-1}\!\left(\frac{1}{7}\right).$$

The other chord is said to subtend an angle

$$\theta_2=\sec^{-1}(7).$$

Now, by definition of the inverse secant,

$$\sec^{-1}(7)=\cos^{-1}\!\left(\frac{1}{7}\right).$$

Thus

$$\theta_2=\theta_1.$$

So both chords actually subtend the same angle, which we denote simply by

$$\theta=\cos^{-1}\!\left(\frac{1}{7}\right).$$

Even though the two chords are on opposite sides of the centre, they are at equal perpendicular distances from the centre O; one distance is measured in one direction, the other in the opposite direction.

For any chord of a circle we have the standard result:

Distance from the centre to the chord $$d=R\cos\!\left(\frac{\text{angle subtended at centre}}{2}\right).$$

Here the angle subtended is $$\theta.$$ Thus the distance of each chord from the centre is

$$d=R\cos\!\left(\frac{\theta}{2}\right).$$

We first evaluate $$\cos\!\left(\dfrac{\theta}{2}\right).$$ Using the half-angle identity

$$\cos\!\left(\frac{\theta}{2}\right)=\sqrt{\frac{1+\cos\theta}{2}},$$

and substituting $$\cos\theta=\frac{1}{7},$$ we get

$$\cos\!\left(\frac{\theta}{2}\right)=\sqrt{\frac{1+\dfrac{1}{7}}{2}} =\sqrt{\frac{\dfrac{8}{7}}{2}} =\sqrt{\frac{8}{14}} =\sqrt{\frac{4}{7}} =\frac{2}{\sqrt7}.$$

Now we substitute this value into the formula for $$d$$:

$$d=R\cos\!\left(\frac{\theta}{2}\right) =2\left(\frac{2}{\sqrt7}\right) =\frac{4}{\sqrt7}.$$

Since the chords are on opposite sides of the centre, the total distance between them is the sum of the two equal perpendicular distances, namely

$$\text{distance between chords}=d+d=2d =2\left(\frac{4}{\sqrt7}\right) =\frac{8}{\sqrt7}.$$

Among the given alternatives this matches Option A.

Hence, the correct answer is Option A.