What are the coordinates of the center of the Sun if the center of the hyperbolic orbit is​ (0,0) and the Sun lies on the positive​ x-axis?

Question

A comet following a hyperbolic path about the Sun has a perihelion of 180 Gm. When the line from the comet to the Sun is perpendicular to the focal axis of the orbit, the comet is 562.5 Gm from the Sun. Calculate a, b, c, and e. What are the coordinates of the center of the Sun if the center of the hyperbolic orbit is (0,0) and the Sun lies on the positive x-axis?

Answer 1

The distance 562.5 Gm is called the semilatus rectum.   In terms of the usual parameters the semilatus rectum = b²/a

The perihelion   is   sqrt(a² + b²) - a    So the equations for a and b are

 

      b²/a  =  562.5

      sqrt( a² + b²) - a   =  180

 

    This set is easily solved by using the first equation to eliminate b² from the second equation  leading to

 

    sqrt( a² + 562.5 a) = 180 + a

 

After squaring both sides and collecting like terms ( the a² cancels out)   the result for a is

  a = 160   Gm

  substituting back into the first equation gives  b = 300 Gm

  c = sqrt( a² + b²) =   340 Gm   This is also the x coordinate of the sun.

   So the coordinates of the sun are (340, 0)