(Solved): FOCUS-DIRECTRIX DERIVATION OF ELLIPSE An ellipse is the locus of points in a plane such that the s ...

FOCUS-DIRECTRIX DERIVATION OF ELLIPSE An ellipse is the locus of points in a plane such that the sum of the distances from two fixed points is constant. Each fixed point is called a focus. The standard form for an ellipse with horizontal orientation centered at the origin is given by x² y² + = 1 a² 6² The derivation of this formula comes from using the locus definition and applying the distance formula. A step- by-step proof can be found on Wolfram MathWorld. Some important points of the proof include the designation of the constant sum as 2a and the coordinates of the foci as (±c, 0). Furthermore, the substitution of a² - c² with 6² greatly simplifies the standard form. In class, we discussed the connection between an ellipse and its directrix. The following is an alternate definition of an ellipse. An ellipse is the locus of points in a plane such that the ratio of the distance to a fixed point and the distance to a fixed line is constant and between 0 and 1. The fixed point is the focus, the fixed line is the directrix, and the constant ratio is the eccentricity. Using this definition, derive the standard form of the equation of an ellipse. For the derivation, assume that the ellipse is centered at the origin, the focus has coordinates (c, 0), where c> 0; the directrix has equation x = a, where 0 < c < a; and the constant ratio is . Clearly indicate which expressions represent the distances between a point in the locus to the focus and to the directrix. A thorough proof will earn 4 extra credit points in the exam category. Submit your completely legible work by Friday, June 10, 2022.