1. The equation of the hyperbola is:
\[ \frac{x^2}{4} - \frac{y^2}{9} = 1. \]
2. The equation of the chord parallel to \(y = 2x\) is:
\[ y = 2x + c. \]
3. Using the midpoint formula for a hyperbola: - The midpoint satisfies the locus equation derived from substituting \(y = 2x + c\) into the hyperbola equation.
4. After simplification, the locus of the midpoint is:
\[ 9x - 8y = 0. \]
Let $A$ and $B$ be two distinct points on the line $L: \frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2}$. Both $A$ and $B$ are at a distance $2\sqrt{17}$ from the foot of perpendicular drawn from the point $(1, 2, 3)$ on the line $L$. If $O$ is the origin, then $\overrightarrow{OA} \cdot \overrightarrow{OB}$ is equal to:
Let the shortest distance between the lines $\frac{x-3}{3} = \frac{y-\alpha}{-1} = \frac{z-3}{1}$ and $\frac{x+3}{-3} = \frac{y+7}{2} = \frac{z-\beta}{4}$ be $3\sqrt{30}$. Then the positive value of $5\alpha + \beta$ is