Coordinate Geometry
Hyperbola
Grade Class 12
Question:
On the hyperbola $y^2 - x^2 = 1$, consider a point $P$ with abscissa $n$ (integer). Let $d_n$ be the shortest distance from $P$ to the line $y = x$. Then $\lim_{n \to \infty} n \cdot d_n$ equals
Step-by-Step Solution
Key Concept: Distance from $(n, \sqrt{1+n^2})$ to $y=x$ is $|\sqrt{1+n^2}-n|/\sqrt{2}$; multiply by $n$ and take limit.
$d_n = \frac{|\sqrt{1+n^2} - n|}{\sqrt{2}} = \frac{1}{\sqrt{2}(\sqrt{1+n^2}+n)}$. So $n\cdot d_n = \frac{n}{\sqrt{2}(\sqrt{1+n^2}+n)} \to \frac{1}{2\sqrt{2}}$ as $n\to\infty$.
Correct Answer: 1