Tangents are drawn to the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$, parallel to the line $2x - y = 1$. The points of contact of the tangents on the hyperbola are _____ (give both points).
Step-by-Step Solution
Key Concept: Slope $m = 2$; tangent $y = 2x + c$ with $c^2 = 9(4) - 4 = 32, c = \pm 4\sqrt{2}$. Point of contact: $(a^2m/c, -b^2/c) = (9 \cdot 2/c, -4/c)$. For $c = 4\sqrt{2}$: $(9/(2\sqrt{2}), -1/\sqrt{2}) = (9\sqrt{2}/4, -\sqrt{2}/2)$. For $c = -4\sqrt{2}$: $(-9\sqrt{2}/4, \sqrt{2}/2)$.
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Correct Answer: (\frac{9\sqrt{2}}{4}, -\frac{\sqrt{2}}{2}), (-\frac{9\sqrt{2}}{4}, \frac{\sqrt{2}}{2})