Given the equation x
^{2}/9 + y
^{2}/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 x^{2}/9  y^{2}/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 20111124 14:28:58 