if ` x = a cos^3 theta sin^2 theta and y = a cos^2 theta sin^3 theta` and `(x^2 + y^2)^p/(xy)^q` is independent of `theta`, then (A) `4p=5q` (B) `5p=4q` (C) `p+q=9` (D) `pq=20`

Options

A. 4P=5Q
B. 5P=4Q
C. `P+Q=9
D. PQ=20

Answer

To solve the problem, we start with the given expressions for \( x \) and \( y \):

\[ x = a \cos^3 \theta \sin^2 \theta \]
\[ y = a \cos^2 \theta \sin^3 \theta \]

### Step 1: Calculate \( x^2 + y^2 \)

First, we calculate \( x^2 \) and \( y^2 \):

\[ x^2 = (a \cos^3 \theta \sin^2 \theta)^2 = a^2 \cos^6 \theta \sin^4 \theta \]
\[ y^2 = (a \cos^2 \theta \sin^3 \theta)^2 = a^2 \cos^4 \theta \sin^6 \theta \]

Now, we can add \( x^2 \) and \( y^2 \):

\[ x^2 + y^2 = a^2 \cos^6 \theta \sin^4 \theta + a^2 \cos^4 \theta \sin^6 \theta \]

Factoring out \( a^2 \cos^4 \theta \sin^4 \theta \):

\[ x^2 + y^2 = a^2 \cos^4 \theta \sin^4 \theta \left( \cos^2 \theta + \sin^2 \theta \right) \]

Using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \):

\[ x^2 + y^2 = a^2 \cos^4 \theta \sin^4 \theta \]

### Step 2: Calculate \( xy \)

Next, we calculate \( xy \):

\[ xy = (a \cos^3 \theta \sin^2 \theta)(a \cos^2 \theta \sin^3 \theta) = a^2 \cos^5 \theta \sin^5 \theta \]

### Step 3: Substitute into the expression

We need to evaluate the expression:

\[ \frac{(x^2 + y^2)^p}{(xy)^q} \]

Substituting the values we found:

\[ \frac{(a^2 \cos^4 \theta \sin^4 \theta)^p}{(a^2 \cos^5 \theta \sin^5 \theta)^q} \]

This simplifies to:

\[ \frac{a^{2p} \cos^{4p} \theta \sin^{4p} \theta}{a^{2q} \cos^{5q} \theta \sin^{5q} \theta} \]

### Step 4: Combine the terms

This can be rewritten as:

\[ a^{2(p - q)} \cos^{4p - 5q} \theta \sin^{4p - 5q} \theta \]

### Step 5: Set the expression independent of \( \theta \)

For the expression to be independent of \( \theta \), both the powers of \( \cos \theta \) and \( \sin \theta \) must be zero:

1. \( 4p - 5q = 0 \)
2. \( 4p - 5q = 0 \)

This gives us:

\[ 4p = 5q \]

### Conclusion

Thus, the condition for the expression to be independent of \( \theta \) is:

\[ \boxed{4p = 5q} \]