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 conjugate hyperbolas, the eccentricities satisfy $\frac{1}{e^2}+\frac{1}{e'^2}=1$.
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