Proof: An Ellipse with Normal Lines Passing Through Its Center Must Be a Circle

In this article, we will present a geometric proof that an ellipse with the property that its normal lines pass through its center must actually be a circle. This involves understanding the properties of ellipses, their normal lines, and the mathematical relationships between them.

Definitions and Setup

Consider an ellipse defined by the equation:

x2/a2 y2/b2 1

where a and b are the semi-major and semi-minor axes, respectively. The center of the ellipse is at the origin (0, 0).

Normal Line to the Ellipse

To find the normal line to the ellipse at a point (x?, y?) on the ellipse, we first need to find the slope of the tangent line at that point. The parametric equations of the ellipse are:

x a cos θ and y b sin θ

The derivatives with respect to θ give us the components of the tangent vector:

dx/dθ - a sin θ and dy/dθ b cos θ

The slope of the tangent line at (x?, y?) is:

slope of tangent (dy/dθ) / (dx/dθ) -b/a cot θ

Thus, the slope of the normal line, which is perpendicular to the tangent, is:

slope of normal 1/(-b/a cot θ) a/b tan θ

Equation of the Normal Line

The equation of the normal line at the point (x?, y?) is given by:

y - y0 (a/b) tan θ (x - x0)

Substituting x0 a cos θ and y0 b sin θ:

y - b sin θ (a/b) tan θ x - a cos θ

Condition for the Normal Line to Pass Through the Center

For the normal line to pass through the center (0, 0), we can substitute (0, 0) into the normal line equation:

0 - b sin θ (a/b) tan θ 0 - a cos θ

This simplifies to:

-b sin θ -(a2/b) tan θ cos θ

Using tan θ (sin θ / cos θ), we have:

-b sin θ -(a2/b) (sin θ / cos θ) cos θ

This further simplifies to:

-b sin θ -(a2/b) sin θ

Assuming sin θ ≠ 0 (which is true for points on the ellipse except the endpoints of the major and minor axes), we can divide both sides by sin θ:

-b -(a2/b)

Multiplying through by -b gives:

b2 a2

Conclusion

The condition b2 a2 implies that a b. Therefore, the ellipse is a circle with radius a (or b, since a and b are equal).

Thus, we have shown that if the normal line to each point on an ellipse passes through the center of the ellipse, then the ellipse must be a circle.

Final Result

If the normal to each point on an ellipse passes through its center, then the ellipse is a circle.