Equation of Parabola in Reduced Form/Cartesian Frame

Theorem

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

That is:

$(1)$ its focus is at the point $\tuple {c, 0}$

$(2)$ its directrix is aligned with the line $x = -c$

for some $c \in \R_{> 0}$.

The equation of $K$ is:

$y^2 = 4 a x$

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): parabola
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): parabola

• Weisstein, Eric W. "Parabola." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Parabola.html