Parabola

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The standard equation of a parabola with vertex at origin and focus on positive x-axis is

RankGuru · Accepted Answer

1. **Locus Definition**: A parabola is the set of points equidistant from a focus $(a, 0)$ and a directrix $x = -a$.
2. **Derivation**: For a point $(x, y)$, $\sqrt{(x-a)^2 + y^2} = |x+a|$.
3. **Simplification**: $(x-a)^2 + y^2 = (x+a)^2 \implies x^2 - 2ax + a^2 + y^2 = x^2 + 2ax + a^2$.
4. **Result**: $y^2 = 4ax$. This parabola opens to the right and is symmetric about the x-axis.

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