Finding the Tangents from a Point to an Ellipse Exploiting its Reflective Property

To find the equations of tangent lines to an ellipse is traditionally considered a complex task, often requiring advanced mathematical skills such as calculus and trigonometry. This paper presents a new demonstration of an overlooked approach that simplifies this problem considerably. In particular, a system of three equation to represent the angular coefficient of the tangents was solved. The proof leverages the Reflective Property of the ellipse, and it employs it in a manner that extends beyond traditional geometric applications. The demonstrated method is versatile; it is applicable in various scenarios, whether the known point lies on the curve or the ellipse is translated. Finally, the integration of this approach into high-school mathematics education is recommended to provide students with a more accessible means of solving this challenging problem - the described technique requires neither long calculations nor the understanding of the aforementioned branches of mathematics.