Equation of Ellipse in Reduced Form/Cartesian Frame/Proof 2

Proof

 * EllipseEquation-2.png

Let $P$ be an arbitrary point in the plane.

Let $PM$ be dropped perpendicular to $V_1 V_2$.

Hence $M = \tuple {x, 0}$.

From Intersecting Chord Theorem for Conic Sections:
 * $PM^2 = k V_1 M \times M V_2$

for some constant $k$.

Hence:

Putting $x = 0$, we find the points where $K$ crosses the $y$-axis are defined by:


 * $y^2 = k a^2$

This gives us:
 * $k a^2 = b^2$

and so:

Hence the result.