Intersection of a Rectangular Hyperbola with a Circle

Understanding the intersection of a rectangular hyperbola with a circle involves analyzing the points at which these two conic sections meet. This topic is significant in coordinate geometry and can be approached by solving the equations of the hyperbola and the circle simultaneously.

Rectangular Hyperbola

A rectangular hyperbola is a specific type of hyperbola where the asymptotes are perpendicular to each other, and hence the transverse and conjugate axes are equal in length. The standard equation of a rectangular hyperbola centered at the origin with axes along the coordinate axes is:

$$ xy = c^2 $$

where $c$ is a real number.

Circle

A circle is a set of all points in a plane that are at a fixed distance (radius) from a fixed point (center). The standard equation of a circle with center at the origin and radius $r$ is:

$$ x^2 + y^2 = r^2 $$

Intersection Points

To find the intersection points of a rectangular hyperbola and a circle, we need to solve their equations simultaneously. This can be done by substituting $y$ from the hyperbola's equation into the circle's equation or vice versa.

Example

Let's consider a rectangular hyperbola $xy = 4$ and a circle $x^2 + y^2 = 16$. To find their intersection points, we can substitute $y = \frac{4}{x}$ from the hyperbola's equation into the circle's equation:

$$ x^2 + \left(\frac{4}{x}\right)^2 = 16 $$

Simplifying, we get:

$$ x^4 - 16x^2 + 16 = 0 $$

Factoring, we have:

$$ (x^2 - 8x + 4)(x^2 + 8x + 4) = 0 $$

Solving for $x$, we get four possible values for $x$: $2\sqrt{2}, -2\sqrt{2}, 2\sqrt{2}, -2\sqrt{2}$. Substituting these back into the hyperbola's equation, we get the corresponding $y$ values.

The intersection points are:

$$ (2\sqrt{2}, 2\sqrt{2}), (-2\sqrt{2}, -2\sqrt{2}), (2\sqrt{2}, -2\sqrt{2}), (-2\sqrt{2}, 2\sqrt{2}) $$

Table of Differences and Important Points

Feature	Rectangular Hyperbola	Circle
Equation	$xy = c^2$	$x^2 + y^2 = r^2$
Axes	Transverse and conjugate axes are equal	Radius is the same in all directions
Asymptotes	Perpendicular to each other	No asymptotes
Symmetry	Central symmetry about the origin	Radial symmetry about the center
Intersection Points	Found by substitution and solving a polynomial equation	Found by substitution and solving a quadratic equation

Formulas

For a rectangular hyperbola $xy = c^2$ and a circle $x^2 + y^2 = r^2$, the intersection points can be found by solving:

$$ x^2 + \left(\frac{c^2}{x}\right)^2 = r^2 $$

which simplifies to a quartic equation:

$$ x^4 - r^2x^2 + c^4 = 0 $$

Conclusion

The intersection of a rectangular hyperbola with a circle can be determined by solving their equations simultaneously. This involves algebraic manipulation and solving polynomial equations. The intersection points are crucial in various applications, including physics, engineering, and computer graphics. Understanding this concept requires a solid foundation in algebra and geometry.