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The Best Explanation of the Equation of an Ellipse
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The Best Explanation of the Equation of an Ellipse

Deriving the Equation of a Horizontal Ellipse

In this video, we will learn about deriving the equation of a horizontal ellipse. We will explore the concept of foci, major and minor axes, and use algebraic steps to simplify the equation.

Understanding Ellipses

  • An ellipse is created by connecting two points called foci with a string.
  • The length of the string remains constant, creating an ellipse when extended around the foci.
  • The longest diameter of the ellipse is called the major axis, while the shortest diameter is called the minor axis.
  • Let's denote half the length of the major axis as 'a' and half the length of the minor axis as 'b'.

Pythagorean Theorem and Distance Formula

  • By applying Pythagorean theorem to a triangle formed by extending the string evenly at the top, we have B^2 + C^2 = A^2.
  • We label distances from each focus to an XY coordinate as D1 and D2.
  • Using distance formula, we can find D1 and D2 by substituting coordinates into the formula.

Simplifying Equations

  • After simplifying equations and rearranging terms, we square both sides to eliminate square roots.
  • Expanding parentheses and canceling out like terms helps us further simplify equations.
  • Dividing both sides by A^2 * (A^2 - C^2), we eliminate common factors.

Equation of a Horizontal Ellipse

  • Finally, after factoring and substituting B^2 = A^2 - C^2 into our equation, we obtain x^2 / A^2 + y^2 / (A^2 - C^2) = 1.
  • This represents all points (x,y) that make up a horizontal ellipse.

Thank you for watching this video on deriving the equation of a horizontal ellipse.

Video description

I made this with a lot of heart, and every purchase helps me keep creating. If you like what I do or just want to support independent makers, check it out: http://amazon.com/gp/product/B093D22DGN Thank you for being here—it really means a lot. In this video we derive the equation of an ellipse. An ellipse is defined as the locus of all points in the plane for which the sum of the distances r1 and r2 to two fixed points F1 and F2 (called the foci) separated by a distance 2c, is a given constant 2a. Therefore, from this definition the equation of the ellipse is: r1 + r2 = 2a, where a = semi-major axis.