Let P(2,1) be a point in circle C with center at (0,0) and a radius of 3. For every point X on circle C, a perpendicular straight line is drawn through the midpoint of XP. The envelope of the family of all these straight lines is an ellipse. Find the equation of this ellipse.
Insight 1: The construction is essentially a variant of Newton's Ellipse, so O, the centre of C, and P must be the foci of the ellipse.
Insight 2: Consider ray OP cutting C at X. Let the mid point of XP be Y. The perpendicular here must be tangent to the ellipse, so Y is on the ellipse.
The equation of the ellipse is then 5x^24xy+8y^28x4y=4
Edited on May 7, 2020, 11:34 pm

Posted by broll
on 20200507 23:19:44 