Let $P(x, y)$ is a variable point such that $\sqrt{(x-1)^2 + (y-2)^2} - \sqrt{(x-5)^2 + (y-5)^2} = 3$ which represents hyperbola. The eccentricity $e'$ of the corresponding conjugate hyperbola is:
Step-by-Step Solution
Key Concept: For a hyperbola defined by ||PF₁| - |PF₂|| = 2a, identify that 2a = 3 (not 5), then use the distance between foci (2c = 5) to find e = c/a = 5/3. Apply the conjugate hyperbola eccentricity relation 1/e² + 1/e'² = 1 to obtain e' = 5/4.
Given foci at $(1,2)$ and $(5,5)$ with distance $5$ between them, we have $2a=5$ so $2ae=5$ and $e=\frac{5}{3}$. For the conjugate hyperbola, the eccentricity relation $\frac{1}{e^2}+\frac{1}{e'^2}=1$ yields $e'=\frac{5}{4}$ for the corresponding conjugate hyperbola.
Correct Answer: 3