In this problem, we are given the equation:

\(p \sin^2 \alpha + q \cos^2 \alpha = m\)

We need to find the value of \(\tan^2\alpha\) using the given condition.

Let's start by manipulating the given equation:

1. Use the identity \(\sin^2 \alpha + \cos^2 \alpha = 1\).
2. Express \(\sin^2 \alpha\) in terms of \(\cos^2 \alpha\):
   \(\sin^2 \alpha = 1 - \cos^2 \alpha\).
3. Substitute \(\sin^2 \alpha\) in the given equation:
   \(p(1 - \cos^2 \alpha) + q \cos^2 \alpha = m\)
4. Simplify it:
   \(p - p \cos^2 \alpha + q \cos^2 \alpha = m\)
   \(p + (q - p)\cos^2 \alpha = m\)
5. Isolating \(\cos^2 \alpha\), we get:
   \((q - p) \cos^2 \alpha = m - p\)
   \(\cos^2 \alpha = \frac{m - p}{q - p}\)
6. Using \(\tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha}\), we can express it as:
   \(\tan^2 \alpha = \frac{1 - \cos^2 \alpha}{\cos^2 \alpha}\)
    
7. Now substitute for \(\cos^2 \alpha\) in \(\tan^2 \alpha\):
   \(\tan^2 \alpha = \frac{1 - \frac{m-p}{q-p}}{\frac{m-p}{q-p}}\)
8. Simplifying the expression:
   \(\tan^2 \alpha = \frac{(q-p) - (m-p)}{m-p}\)
   \(\tan^2 \alpha = \frac{q-m}{m-p}\)

Thus, the correct expression for \(\tan^2 \alpha\) is \(\frac{m-q}{p-m}\).