This problem requires us to find a specific point on a hyperbola, $(x_1, y_1)$, where the tangent line is parallel to a given straight line. Once we determine this point, we need to evaluate a given algebraic expression involving $x_1$ and $y_1$. This involves understanding the standard forms of hyperbola and its tangent, as well as properties of parallel lines.

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Key Concepts and Formulas

1. Standard Equation of a Hyperbola: A hyperbola centered at the origin has the equation: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ For the given hyperbola $\frac{x^2}{4} - \frac{y^2}{2} = 1$, we can identify $a^2 = 4$ and $b^2 = 2$.

2. Equation of a Tangent to a Hyperbola at a Point $(x_1, y_1)$: If a point $(x_1, y_1)$ lies on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, then the equation of the tangent line to the hyperbola at this point is given by: $\frac{x x_1}{a^2} - \frac{y y_1}{b^2} = 1$ This formula is derived using differentiation or by the T-substitution method for conic sections.

3. Slope of a Line: For a linear equation in the form $Ax + By + C = 0$, the slope $m$ is given by $m = -\frac{A}{B}$. Alternatively, by rewriting it in the slope-intercept form $y = mx + c$, the slope is $m$.

4. Parallel Lines: Two non-vertical lines are parallel if and only if they have the same slope.

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Step-by-Step Solution

Step 1: Determine the slope of the given line.

• Explanation: The problem states that the tangent line to the hyperbola is parallel to the straight line $2x - y = 0$. A fundamental property of parallel lines is that they share the same slope. Therefore, our first step is to calculate the slope of the given line, which will also be the slope of our desired tangent line.
• The given line is $2x - y = 0$.
• To find its slope, we can rearrange the equation into the slope-intercept form ($y = mx + c$). $y = 2x$