Equation of Tangent 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 is given by:

$$ xy = c^2 $$

where ( c ) is a constant.

Tangent to a Rectangular Hyperbola

The equation of the tangent to a rectangular hyperbola at a point ( (x_1, y_1) ) on the hyperbola can be derived using the concept of the slope or by using implicit differentiation.

Using Implicit Differentiation

Given the standard equation of the rectangular hyperbola ( xy = c^2 ), we differentiate both sides with respect to ( x ):

$$ \frac{d}{dx}(xy) = \frac{d}{dx}(c^2) $$

This gives us:

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

Solving for ( \frac{dy}{dx} ), we get the slope of the tangent at any point ( (x, y) ):

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

The equation of the tangent line at ( (x_1, y_1) ) is then:

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

Simplifying, we get the equation of the tangent:

$$ xx_1 + yy_1 = c^2 $$

Using the Slope Form

Alternatively, the equation of the tangent to the rectangular hyperbola can be written in terms of the slope ( m ) as:

$$ y = mx \pm c\sqrt{1 + m^2} $$

where ( m ) is the slope of the tangent line.

Table of Differences and Important Points

Feature	Rectangular Hyperbola	General Hyperbola
Equation	( xy = c^2 )	( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 ) or ( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 )
Asymptotes	Perpendicular to each other	Not necessarily perpendicular
Axes Lengths	Equal (transverse = conjugate)	Unequal
Tangent Equation (Point Form)	( xx_1 + yy_1 = c^2 )	( \frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1 )
Tangent Equation (Slope Form)	( y = mx \pm c\sqrt{1 + m^2} )	( y = mx \pm \sqrt{a^2m^2 + b^2} )

Examples

Example 1: Point Form

Find the equation of the tangent to the rectangular hyperbola ( xy = 9 ) at the point ( (3, 3) ).

Solution:

Using the point form of the tangent equation:

$$ xx_1 + yy_1 = c^2 $$

we substitute ( x_1 = 3 ), ( y_1 = 3 ), and ( c^2 = 9 ):

$$ x(3) + y(3) = 9 $$

Simplifying, we get:

$$ x + y = 3 $$

This is the equation of the tangent to the rectangular hyperbola at the point ( (3, 3) ).

Example 2: Slope Form

Find the equation of the tangent to the rectangular hyperbola ( xy = 16 ) with a slope of ( 2 ).

Solution:

Using the slope form of the tangent equation:

$$ y = mx \pm c\sqrt{1 + m^2} $$

we substitute ( m = 2 ) and ( c^2 = 16 ) (so ( c = 4 )):

$$ y = 2x \pm 4\sqrt{1 + 2^2} $$

Simplifying, we get:

$$ y = 2x \pm 4\sqrt{5} $$

So there are two possible tangents with a slope of ( 2 ), depending on the sign chosen for ( \pm ):

$$ y = 2x + 4\sqrt{5} $$ $$ y = 2x - 4\sqrt{5} $$

These are the equations of the tangents to the rectangular hyperbola with a slope of ( 2 ).