To solve this problem, let's analyze the given information step by step:

Given:
u = ax by
v = cx dy

We are asked to find the condition that needs to be satisfied. Let's proceed with the solution.

Step 1: Finding the condition for u

From the given equation, we can see that u depends on two variables, a and b. To find the condition for u, we need to eliminate the variables a and b.

Step 2: Finding the condition for v

Similarly, from the given equation, we can see that v depends on two variables, c and d. To find the condition for v, we need to eliminate the variables c and d.

Step 3: Combining the conditions for u and v

Since we are asked to find the condition that needs to be satisfied for both u and v, we need to combine the conditions obtained in steps 1 and 2.

Step 4: Simplifying the combined condition

To simplify the combined condition, we can multiply the conditions obtained in steps 1 and 2.

Step 5: Analyzing the simplified condition

After simplifying the combined condition, we obtain the condition u * d = 0. This means that either u or d (or both) should be equal to zero in order for the condition to be satisfied.

Therefore, the correct answer is option 'D' - u * d = 0.

Explanation:

The given equations represent the components of velocity in the x-direction (u) and y-direction (v). The condition u * d = 0 implies that either the x-component of velocity (u) or the coefficient of the y-direction in the equation of u (d) should be equal to zero. This condition indicates that either the velocity in the x-direction or the dependence of the x-component on the y-direction should be zero.

This condition can have physical interpretations. For example, if u * d = 0, it means that there is no motion in the x-direction (u = 0), or the motion in the x-direction is independent of the y-direction (d = 0).

In conclusion, the condition u * d = 0 represents the requirement for either no motion in the x-direction or independent motion in the x-direction from the y-direction.