The Locus of the Foot of Perpendicular from the Center of an Ellipse on Any Tangent Line

This article delves into the mathematical concept of the locus of the foot of perpendicular drawn from the center of an ellipse onto any tangent line. The inquiry corrects an earlier misconception and provides a comprehensive explanation supported by mathematical derivations and diagrams, unraveling the complex yet fascinating geometric relationships governing ellipses.

Introduction to Ellipses

The equation of a standard ellipse centered at the origin is defined as:

.multipart equation [ frac{x^2}{a^2} frac{y^2}{b^2} 1 ]

where a and b are the semi-major and semi-minor axes, respectively. The foci of this ellipse are located at (±ae, 0), where e is the eccentricity of the ellipse.

Tangent Lines to an Ellipse

An important property of an ellipse is that a tangent line to the ellipse at any point can be determined. The equation of a tangent line at a point (x1, y1) on the ellipse can be given by:

multipart equation [ frac{x x_1}{a^2} frac{y y_1}{b^2} 1 ]

However, for our purpose, we consider the general form of a tangent line to the ellipse, which can be represented as:

multipart equation [ y mx sqrt{a^2 m^2 b^2} ]

where m is the slope of the tangent line.

Finding the Foot of Perpendicular

The goal is to find the foot of the perpendicular from the origin (0, 0) to this tangent line. The foot of the perpendicular is the point where a line, perpendicular to the tangent, intersects the tangent. If the equation of the line perpendicular to the tangent from the origin is:

multipart equation [ y -frac{1}{m}x ]

Solving the system of these two equations, we find the coordinates of the foot of the perpendicular.

Derivation

Solving the system of equations:

multipart equation [ y mx sqrt{a^2 m^2 b^2} ] [ y -frac{1}{m}x ]

Substitute ( y -frac{1}{m}x ) into the first equation:

multipart equation [ -frac{1}{m}x mx sqrt{a^2 m^2 b^2} ] [ -frac{x}{m} - mx sqrt{a^2 m^2 b^2} ] [ x(-frac{1}{m} - m) sqrt{a^2 m^2 b^2} ] [ x -frac{m sqrt{a^2 m^2 b^2}}{1 m^2} ]

Substitute this value of ( x ) back into ( y -frac{1}{m}x ) to find ( y ):

multipart equation [ y -frac{1}{m} left( -frac{m sqrt{a^2 m^2 b^2}}{1 m^2} right) ] [ y frac{sqrt{a^2 m^2 b^2}}{1 m^2} ]

Locus of the Foot of Perpendicular

The locus of the foot of the perpendicular for all possible values of m (i.e., slopes of tangents) is a curve that can be described mathematically. This curve is bi-symmetrical and double-lobed, meeting the ellipse at the ends of the major and minor axes. In the special case of a circle (where (a b)), the locus of the foot of the perpendicular is the circle itself.

The exact shape of this curve varies depending on the eccentricity of the ellipse. For a general ellipse, the locus of the foot of the perpendicular is a curve defined by the equations derived above, which can be visualized as:

multipart equation [ left( frac{x}{a} right)^2 left( frac{y}{b} right)^2 1 - left[ frac{x sqrt{a^2 m^2 b^2}}{m (1 m^2)} right]^2 ]

Conclusion

The locus of the foot of the perpendicular drawn from the center of an ellipse to any of its tangents is a fascinating and intricate geometric figure. This curve is not just a point but a continuous set of points that define a specific shape based on the properties of the ellipse itself. Understanding this concept provides valuable insights into the geometric relationships inherent in conic sections.

Keywords: elliptical locus, foot of perpendicular, tangent line, ellipse equation