Question

If \( x = p \cos^3 \alpha \) and \( y = q \sin^3 \alpha \), then the value of
\( \left( \frac{x}{p} \right)^{2/3} + \left( \frac{y}{q} \right)^{2/3} \text{ is:} \)

• A. 1
• B. 2
• C. p
• D. q

Hint:

In problems with powers like \( \cos^3 \alpha \) or \( \sin^3 \alpha \), apply the inverse power operation carefully. Use identities like \( \sin^2 \alpha + \cos^2 \alpha = 1 \) to simplify expressions.

Answer 1

The Correct Option is A

We are given: \[ x = p \cos^3 \alpha \quad \text{and} \quad y = q \sin^3 \alpha \] Step 1: Divide both sides of the equation \( x = p \cos^3 \alpha \) by \( p \): \[ \frac{x}{p} = \cos^3 \alpha \] Step 2: Raise both sides to the power \( \frac{2}{3} \): \[ \left( \frac{x}{p} \right)^{2/3} = (\cos^3 \alpha)^{2/3} = \cos^2 \alpha \] Step 3: Similarly, divide both sides of the second equation \( y = q \sin^3 \alpha \) by \( q \): \[ \frac{y}{q} = \sin^3 \alpha \] Step 4: Raise both sides to the power \( \frac{2}{3} \): \[ \left( \frac{y}{q} \right)^{2/3} = (\sin^3 \alpha)^{2/3} = \sin^2 \alpha \] Step 5: Add the two results: \[ \left( \frac{x}{p} \right)^{2/3} + \left( \frac{y}{q} \right)^{2/3} = \cos^2 \alpha + \sin^2 \alpha \] Step 6: Apply the Pythagorean identity: \[ \cos^2 \alpha + \sin^2 \alpha = 1 \] So, the final answer is: \[ \boxed{1} \]