Exploring New Equations in Conic Sections: A Creative Journey
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Chapter 1: The World of Conic Sections
My explorations have led me to the intriguing realm of conic sections, a foundational area in mathematics. Surprisingly, this field is often overlooked when it comes to discovering new formulas, as it has been a staple of mathematics education for centuries. For instance, Fermat's Last Theorem remained an unsolved mystery for over 300 years until Andrew Wiles provided a solution, yet its complexity makes it inaccessible to many.
The 1670 edition of Diophantus’s Arithmetica contains Fermat’s commentary on what is now known as his "Last Theorem," published posthumously by his son. This historical context highlights the depth of mathematical inquiry.
Section 1.1: Rethinking Basic Concepts
Consider how educators introduce lines and slopes in their curriculum. Soon after, they delve into parabolas and cover essential formulas such as those for slope and distance. Yet, a crucial question often goes unasked: “What happens if a line intersects the vertex (or focus) of a parabola (or ellipse or hyperbola) at a specific slope and distance?” This inquiry seems to have been overlooked.
Subsection 1.1.1: A Personal Inquiry
I pondered this question regarding all conic sections and their unique points. To my astonishment, I derived several new equations. I've shared some of these findings on Medium, with my latest contribution available for discussion.
Section 1.2: Unveiling New Equations
In my recent work, I introduced three equations that relate to an up-down parabola. These equations are depicted in my latest Medium post. They explore the relationship between the line and the parabola, offering fresh insights into this well-trodden territory.
Chapter 2: Sharing My Findings
I am thrilled to continue sharing my discoveries on Medium. Stay tuned for more insights and equations as I delve deeper into the fascinating world of mathematics. Thank you for following my journey!