To solve this problem, we need to analyze the scenario and calculate the longer distance between points $X$ and $Y$ along the circumference of the circular track.

Step-by-Step Solution

1. Given Information:

   • The circular track has a circumference of 540 miles.
   • Tom's speed = 20 miles/hour, and Harry's speed = 4 miles/hour.
   • They meet for the 17th time at point $X$ and for the 27th time at point $Y$.

2. Relative Speed: Since they are running in opposite directions, their relative speed is the sum of their speeds: $\text{Relative speed} = 20 + 4 = 24 \text{ miles/hour}$

3. Time Between Meetings: The time taken for one meeting (or lap) around the track in terms of their relative speed is: $\text{Time for one meeting} = \frac{\text{Circumference}}{\text{Relative speed}} = \frac{540}{24} = 22.5 \text{ hours}$

4. Distance Covered Between Meetings: Since they meet every 22.5 hours, we calculate the distance they cover along the track for each meeting: $\text{Distance covered in one meeting} = 22.5 \times 20 = 450 \text{ miles (for Tom)}$ This calculation matches Tom's speed multiplied by the time per meeting, considering he goes around the track in steps of 450 miles.

5. Meeting at Points $X$ and $Y$:

   • The 17th meeting at point $X$ means that Tom has covered $17 \times 450$ miles along the track.
   • The 27th meeting at point $Y$ means Tom has covered $27 \times 450$ miles along the track.

6. Difference in Distance: To find the distance between points $X$ and $Y$, calculate the difference in the number of laps: $\text{Difference in meetings} = (27 - 17) \times 450 = 10 \times 450 = 4500 \text{ miles}$