Equidistance of Ellipse equals Major Axis
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Theorem
Let $K$ be an ellipse whose foci are $F_1$ and $F_2$.
Let $P$ be an arbitrary point on $K$.
Let $d$ be the constant distance such that:
- $d_1 + d_2 = d$
where:
- $d_1 = P F_1$
- $d_2 = P F_2$
Then $d$ is equal to the major axis of $K$.
Proof
By the equidistance property of $K$:
- $d_1 + d_2 = d$
applies to all points $P$ on $K$.
Thus it also applies to the two vertices $V_1$ and $V_2$:
- $V_1 F_1 + V_1 F_2 = d$
- $V_2 F_1 + V_2 F_2 = d$
Adding:
- $V_1 F_1 + V_2 F_1 + V_1 F_2 + V_2 F_2 = 2 d$
But:
- $V_1 F_1 + V_2 F_1 = V_1 V_2$
- $V_1 F_2 + V_2 F_2 = V_1 V_2$
and so:
- $2 V_1 V_2 = 2 d$
By definition, the major axis is $V_1 V_2$.
Hence the result.
$\blacksquare$
Sources
- Weisstein, Eric W. "Ellipse." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Ellipse.html