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

Theorem

Let $K$ be an ellipse aligned in a cartesian plane in reduced form.

Let:

the major axis of $K$ have length $2 a$

the minor axis of $K$ have length $2 b$.

The equation of $K$ is:

$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$

Proof

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:

\(\ds y^2\)

\(=\)

\(\ds k \paren {a + x} \paren {a - x}\)

\(\ds \)

\(=\)

\(\ds k \paren {a^2 - x^2}\)

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:

\(\ds y^2\)

\(=\)

\(\ds \dfrac {b^2} {a^2} \paren {a^2 - x^2}\)

\(\ds \)

\(=\)

\(\ds b^2 - x^2 \dfrac {b^2} {a^2}\)

\(\ds \leadsto \ \ \)

\(\ds \dfrac {y^2} {b^2}\)

\(=\)

\(\ds 1 - \dfrac {x^2} {a^2}\)

Hence the result.

$\blacksquare$

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {IV}$. The Ellipse: $2$. To find the equation of the ellipse in its simplest form