To determine the ratio in which shares are issued on a pro-rata basis, let's analyze the given details:

1. Eicher Ltd. received applications for a total of 72,000 shares.
2. The company had issued and allotted 50,000 shares. However, they received applications that totaled 72,000 shares.
3. Allotment was done pro-rata to the applicants applying for 65,000 shares. This means that the total application for 65,000 shares had to be pro-rated to fit into the 50,000 shares available.

The pro-rata ratio can be calculated as the number of shares allotted to the number of shares applied for:

\(\text{Pro-rata ratio} = \frac{\text{Number of shares allotted}}{\text{Number of shares applied for}} = \frac{50,000}{65,000}\)

Now, simplify the fraction:

\(\begin{align*} \frac{50,000}{65,000} &= \frac{500}{650} \\ &= \frac{10}{13} \end{align*}\)

This shows that the shares are issued on a pro-rata basis in the ratio 10:13.

Thus, the correct answer is: 10:13.

Let's briefly justify why other options are incorrect:

• \(50:72\): This ratio does not represent the pro-rata allocation because 65,000 applications were prorated to 50,000 shares, not 72,000.
• \(72:50\): This is an inverse ratio that would indicate more shares were allotted than applied, which is incorrect.
• \(13:10\): This is simply the inverse of the correct answer.