Question

Given two operators \( \oplus \) and \( \odot \) on numbers \( p \text{ and } q \) such that \[ p \oplus q = \frac{p^2 + q^2}{pq} \text{and} p \odot q = \frac{p^2}{q}, \] if \( x \oplus y = 2 \odot 2 \), then \( x = \)

• A. \( \frac{y}{2} \)
• B. \( y \)
• C. \( \frac{3y}{2} \)
• D. \( 2y \)

Hint:

To solve problems with operator equations, substitute the given values and simplify using algebraic manipulations. Check each step to ensure consistency.

Answer 1

The Correct Option is B

We are given the following operations: \[ p \oplus q = \frac{p^2 + q^2}{pq}, p \odot q = \frac{p^2}{q}. \]

Step 1: Calculate \( 2 \odot 2 \). 
Using the definition of the \( \odot \) operation, we get: \[ 2 \odot 2 = \frac{2^2}{2} = \frac{4}{2} = 2. \]

Step 2: Solve the equation \( x \oplus y = 2 \). 
Substitute into the equation for \( x \oplus y \): \[ x \oplus y = \frac{x^2 + y^2}{xy}. \] We are told that \( x \oplus y = 2 \), so we have: \[ \frac{x^2 + y^2}{xy} = 2. \]

Step 3: Solve for \( x \). 
Multiply both sides of the equation by \( xy \): \[ x^2 + y^2 = 2xy. \] Rearranging terms: \[ x^2 - 2xy + y^2 = 0. \] This simplifies to: \[ (x - y)^2 = 0, \] so \( x = y \).

Final Answer: \[ y \]