A chord cut the same branch of a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ in $P, P'$ and the asymptotes in $Q, Q'$, then the value of $(PQ + PQ') - (P'Q' + P'Q)$ is____.
Step-by-Step Solution
Key Concept: The symmetry of midpoints for chords of a hyperbola with respect to asymptotes ensures equal chord segments on opposite sides.
For a chord $y = mx + c$ intersecting a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the intersection points satisfy $\frac{x^2}{a^2} - \frac{(mx+c)^2}{b^2} = 1$. The midpoint of the chord and properties of chord geometry show that for a pair of asymptotes $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 0$, the chord endpoints $(x_1, y_1)$ and $(x_3, y_4)$ satisfy $x_1 + x_3 = x_1 + x_4$ and $y_1 + y_2 = y_3 + y_4$. Using the condition that $RQ = RQ'$, $RP = RP'$, and the derived relations $(PQ - P'Q') + (PQ' - P'Q) = 0$, we establish that $PQ = P'Q'$.
Correct Answer: 0