Given the equation x²/9 + y²/4 = 1 find the set of all points from which the angle of view^* of this ellipse is a right angle. What is the significance of this set of points?

Given the equation x²/9 - y²/4 = 1 find the set of all points from which the angle of view^* of this hyperbola is a right angle. What is the significance of this set of points?

* i.e. displaying a right angle between the two tangents.

For the curve 

    x^2         y^2
   ----- + s * ----- = 1,
    a^2         b^2

the circle

   x^2 + y^2 = a^2 + s * b^2

is the locus of points with
an angle of view of 90 degrees.

Where s = +1 for an ellipse and

      s = -1 for a hyperbola.

This was discovered with a 
combination of algebra and
Geometer's Sketchpad.

Note: The four points (a,b), (-a,b), 
      (-a,-b), and (a,-b) are
      clearly members of the locus 
      for the ellipse.

Note: For the hyperbola - if b >= a,
      then there are no points with
      an angle of view of 90 degrees. 

Posted by Bractals on 2011-11-24 14:28:58