Parabolas

Japanese school year: Math C

What you learn

Learn the geometric definition of a parabola based on a focus and directrix, along with its standard Cartesian equation. Parabolas are widely utilized in engineering for designing reflective antennas and analyzing ballistic trajectories. Prior familiarity with quadratic functions and the coordinate distance formula will provide useful background knowledge.

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Key points

This is the standard equation of a horizontal parabola with its vertex at the origin. The constant $p$ defines the focus $(p, 0)$ and directrix $x = -p$ ; it opens right if $p > 0$ and left if $p < 0$ .

y^2 = 4px \quad (p \neq 0)

This is the standard equation of a vertical parabola with its vertex at the origin. The constant $p$ defines the focus $(0, p)$ and directrix $y = -p$ ; it opens upward if $p > 0$ and downward if $p < 0$ .

x^2 = 4py \quad (p \neq 0)

This formula finds the tangent line at a point $(x_1, y_1)$ on the parabola $y^2 = 4px$ . It is formed by replacing $y^2$ with $y_1 y$ and $2x$ with $x + x_1$ .

y_1 y = 2p(x + x_1)

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Practice 1Coming soon