Transverse and conjugate axes of a rectangular hyperbola are along $X$-axis and $Y$-axis respectively and the distance between the foci is $10\sqrt{14}$. Number of the points $(x, y)$ on the curve such that $x$ and $y$ are positive integers, is equal to ______.
Step-by-Step Solution
Key Concept: For a rectangular hyperbola with transverse axis along X-axis, the eccentricity e = √2 is fixed, and using the relation 2ae = distance between foci gives a² = 175. The hyperbola equation x² - y² = 175 must be factored as (x-y)(x+y) = 175 to find integer solutions by examining divisor pairs of 175 = 5² × 7.
Given $ae = 5\sqrt{14}$ and $e = \sqrt{2}$, we find $a^2 = 175$, so the hyperbola equation is $x^2 - y^2 = 175$ or $(x-y)(x+y) = 175$. Since $175 = 5^2 \cdot 7$ and $x > y$, the factor pairs give three solutions: $x - y = 1, x + y = 175$ yields $x = 88, y = 87$; $x - y = 5, x + y = 35$ yields $x = 20, y = 15$; and $x - y = 7, x + y = 25$ yields $x = 16, y = 9$.
Correct Answer: 3