1. Interpret the condition “touching x-axis at a distance 3 from the origin”

If a circle touches the $x$-axis, then its point of contact lies on the $x$-axis.

Given that this point is at a distance $3$ from the origin, the point of contact must be either

Now look at the options: all have equation of the form

which gives center with $x$-coordinate $3$. So the relevant point of contact is $(3,0)$.

Hence the center must be of the form

and radius must be

because the circle touches the $x$-axis.

2. Use the intercept condition on the y-axis

The length of intercept cut by the circle on the $y$-axis is $2\sqrt{7}$.

For a circle with center $(3,k)$ and radius $r=|k|$, the distance of the center from the $y$-axis is $3$.

So the chord cut by the line $x=0$ has length

Given this equals $2\sqrt{7}$,

which gives

Thus

So the possible circles have centers

with radius $4$.

3. Write equations of the possible circles

Case 1: Center $(3,4)$, radius $4$

Expanding:

This matches Option C.

Case 2: Center $(3,-4)$, radius $4$

Expanding:

This matches Option A.

4. Check the remaining options

Option B:

Compare with

Here,

Center is

and radius

This does touch the $x$-axis, but the intercept on the $y$-axis is

which is not $2\sqrt{7}$. So B is incorrect.

Option D:

Center is

radius is again $\frac{7}{2}$. Its intercept on the $y$-axis is also

not $2\sqrt{7}$. So D is incorrect.

5. Final answer

The correct circles are:

6. Comparison with stored correct answer

Stored correct answer: A, C

My derived answer: A, C

So, the answer agrees with the stored correct answer.