The locus of the midpoints of the chords of the hyperbola $\frac{x^2}{4} - \frac{y^2}{3} = 1$ which are parallel to the line $4x + 8y + 1 = 0$ is:
Step-by-Step Solution
Key Concept: Chord with midpoint $(h, k)$: $T = S_1$, i.e., $\frac{hx}{4} - \frac{ky}{3} = \frac{h^2}{4} - \frac{k^2}{3}$. Slope $= \frac{3h}{4k}$. Parallel to $4x + 8y + 1 = 0$ (slope $-1/2$): $\frac{3h}{4k} = -\frac{1}{2} \Rightarrow 6h = -4k \Rightarrow 3x + 2y = 0...$ recheck: slope of chord $= 3h/(4k) = -1/2$ gives $6h = -4k \Rightarrow 3h + 2k = 0$, locus $3x + 2y = 0$. Pick closest option.
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Correct Answer: (2)