The equation of latus rectum of the rectangular hyperbola \(x y=c^{2}\) is (1) \(x-y=2 c+\sqrt{2} c, c>0\) (2) \(x+y=2 c+2 \sqrt{2} c, c>0\) (3) \(x+y=c+\sqrt{2} c, c>0\) (4) \(x+y=2 \sqrt{2} c\)

In the context of hyperbolas, the **latus rectum** is an important geometric feature. It is a line segment that is parallel to the transverse or conjugate axis of the hyperbola. Specifically, for the rectangular hyperbola given by the equation \(xy = c^2\), the latus rectum's perpendicular distance to the center plays a crucial role.
When you visualize this, imagine a line cutting across the hyperbola symmetrically from both sides. This line is the latus rectum.
The length of the latus rectum is particularly significant in determining various other properties of the hyperbola. It essentially helps us understand how 'flattened' or 'stretched' the hyperbola is in relation to its axes.
For our specific equation \(xy = c^2\), the distance from the center ((0,0)) to the latus rectum along either axis is \ceq 0i.e., the latus rectum occurs at x = y = ±\sqrt(2)ci.e., \ x - y = ±c + √2ci.e., and xi+y = ±2c.

Let's break down the **hyperbola equation** we are working with, which is \(xy = c^2\). This formula defines a rectangular hyperbola because the product of the coordinates (x,y) equals a constant, c².
A hyperbola consists of two separate curves, called branches. The rectangular hyperbola is special because its asymptotes are perpendicular to each other, making the X and Y axes bisect the hyperbola at right angles.
To understand it better: • The equation can be rearranged for further interpretation: x = c²/y or y = c²/x. • When any of the variables (x or y) changes, the other one adjusts to maintain the product constant at c².
This balance explains the hyperbola's shape that stretches infinitely in opposite directions. The origin (0,0) acts as the hyperbola's center, and the hyperbola’s shape remains symmetric about this center.

Understanding the **geometric properties** of a rectangular hyperbola helps us in visualizing and solving equations related to it.
The hyperbola given by \(xy = c^2\) has its center at the origin, (0,0). Its asymptotes are the X and Y axes themselves, making this hyperbola symmetric about both axes.
• **Vertices**: Unlike ellipses, hyperbolas don’t have focal points within their vertices. But for a rectangular hyperbola, the vertices expand symmetrically at coordinates (a, a) and (-a, -a), where a is positive.
• **Asymptotes**: The lines the hyperbola approaches but never touches. For \ xy = c^2 \, the asymptotes are x=0 and y=0, the coordinate axes.
• **Transverse and Conjugate axes**: The distance along the axes from the center defines the length of these axes. In rectangular hyperbolas, these lengths are equal.
The lengths of these axes and distances of the latus rectum reached help in defining the bounding limits and maximum directions of the hyperbola's curves.
Knowing these helps particularly in solving and graphing hyperbole related questions confidently.