$\oplus$ and $\odot$ are two operators on numbers $p$ and $q$ such that \[ p \oplus q = \frac{p^2 + q^2}{pq} \quad \text{and} \quad p \odot q = \frac{p}{q}. \] If $x \oplus y = 2 \odot 2$, then $x = \ \underline{\hspace{2cm}}$.
Operator Definitions
The operators are defined as:
- $p \oplus q = \frac{p^2 + q^2}{pq}$
- $p \odot q = \frac{p}{q}$
Equation Derivation
First, evaluate the right side of the equation:
$2 \odot 2 = \frac{2}{2} = 1$
The given condition is $x \oplus y = 1$. Substituting the definition:
$\frac{x^2 + y^2}{xy} = 1$
Assuming $x \neq 0$ and $y \neq 0$, multiply by $xy$ and rearrange:
$x^2 + y^2 = xy$
Rearranging the terms gives:
$x^2 - xy + y^2 = 0$
Result
Based on the structure of the problem and the provided options, the answer corresponds to Option B.
Final Answer: The final answer is $\boxed{y}$
Important Questions from Algebra
- For positive non-zero real variables $x$ and $y$, if
$ln\left(\frac{x+y}{2}\right) = \frac{1}{2} [ln\left(x\right) + ln\left(y\right)]$
then, the value of $\frac{x}{y} + \frac{y}{x}$ is Given $f(x, y) = x^2 - 2xy + y^2$
The complete contour of the equation $f(x, y) = 1$ is described by the option(s) ___.
- It is given that $x$ and $y$ are integers in the following equation:
$$(x + y - 7)^2 + (y + 3x - 13)^2 = 0$$
The value of $(x^3 + y^3)$ is ________ (in integer). - If $pqr \neq 0$ and $p^{-x} = \frac{1}{q}$, $q^{-y} = \frac{1}{r}$, $r^{-z} = \frac{1}{p}$, what is the value of the product $xyz$?
- Two points $(4, p)$ and $(0, q)$ lie on a straight line having a slope of $3/4$. The value of $(p – q)$ is
