Equation of Normal to Rectangular Hyperbola

A rectangular hyperbola is a type of hyperbola where the transverse and conjugate axes are of equal lengths, which means that the asymptotes are perpendicular to each other. The standard equation of a rectangular hyperbola centered at the origin with axes along the coordinate axes is given by:

$$ xy = c^2 $$

where ( c ) is a constant.

Normal to a Rectangular Hyperbola

The normal to a curve at a given point is a straight line that is perpendicular to the tangent at that point. For a rectangular hyperbola, the equation of the normal can be derived using calculus or by using the properties of the hyperbola.

Derivation Using Calculus

To find the equation of the normal, we first need to find the slope of the tangent to the hyperbola at a given point ((x_1, y_1)). The slope of the tangent is the derivative of ( y ) with respect to ( x ). For the hyperbola ( xy = c^2 ), we can implicitly differentiate both sides with respect to ( x ) to get:

$$ y + x \frac{dy}{dx} = 0 $$

Solving for ( \frac{dy}{dx} ), we get:

$$ \frac{dy}{dx} = -\frac{y}{x} $$

At the point ((x_1, y_1)), the slope of the tangent is:

$$ m_t = -\frac{y_1}{x_1} $$

Since the normal is perpendicular to the tangent, its slope ( m_n ) is the negative reciprocal of ( m_t ):

$$ m_n = \frac{x_1}{y_1} $$

Using the point-slope form of the equation of a line, the equation of the normal at ((x_1, y_1)) is:

$$ y - y_1 = \frac{x_1}{y_1}(x - x_1) $$

Simplifying, we get the equation of the normal:

$$ y = \frac{x_1}{y_1}x - \frac{x_1^2}{y_1} + y_1 $$

Important Points and Differences

Property	Tangent to Rectangular Hyperbola	Normal to Rectangular Hyperbola
Definition	Line that touches the curve at a single point without crossing it.	Line perpendicular to the tangent at the point of contact.
Slope at point ((x_1, y_1))	( m_t = -\frac{y_1}{x_1} )	( m_n = \frac{x_1}{y_1} )
Equation	( y = -\frac{y_1}{x_1}x + \frac{2c^2}{x_1} )	( y = \frac{x_1}{y_1}x - \frac{x_1^2}{y_1} + y_1 )

Example

Let's find the equation of the normal to the rectangular hyperbola ( xy = 4 ) at the point ( (2, 2) ).

1. The slope of the tangent at ( (2, 2) ) is ( m_t = -\frac{2}{2} = -1 ).
2. The slope of the normal is the negative reciprocal of ( m_t ), so ( m_n = \frac{2}{2} = 1 ).
3. Using the point-slope form, the equation of the normal is:

$$ y - 2 = 1(x - 2) $$

Simplifying, we get:

$$ y = x $$

Thus, the equation of the normal to the rectangular hyperbola ( xy = 4 ) at the point ( (2, 2) ) is ( y = x ).

In summary, the normal to a rectangular hyperbola at a given point can be found using the slope of the tangent at that point and then applying the point-slope form of a line. The normal is an important concept in geometry and calculus, as it provides a means to analyze the behavior of curves and their properties.