Introduction to Quaternions

43. 1. IF we define a conic section as "the locus of a point which moves so that its distance from a fixed point bears a constant ratio to its distance from a fixed straight line" (Todhunter, Art. 123), we shall find the equation to be (EX. 5, Art. 35)

  α²ρ² = e²(α² - Sαρ)²    (1), 

When e is less than 1, the curve is the ellipse, a few of whose properties we are about to exhibit.

2. SA, SA' are multiples of α: call one of them xα: then, by equation (1), putting xα for ρ, we get

  x² = e²(1 - x)²;

         e
∴ x = ——————— ,
       1 + e

           e
  x = - ——————— ,
         1 - e