Parabola Equations, Focus, and Directrix

Core Concepts

Derivations from focus-directrix definition, standard forms, and key geometric properties.

Parabola Equations, Focus, and Directrix

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What is the geometric definition of a parabola as a locus of points?

A parabola is the set of all coplanar points $P$ whose distance to a fixed point (the focus) equals their perpendicular distance to a fixed line (the directrix).

For a vertical parabola with focus $(0, p)$ and directrix $y = -p$, what equation expresses the locus definition for any point $P(x, y)$?

$$\sqrt{(x - 0)^2 + (y - p)^2} = |y - (-p)|$$

How do you derive the standard equation $x^2 = 4py$ from $\sqrt{x^2 + (y - p)^2} = |y + p|$?

Square both sides: $$x^2 + (y - p)^2 = (y + p)^2$$ Expand both sides: $$x^2 + y^2 - 2py + p^2 = y^2 + 2py + p^2$$ Subtract $y^2 + p^2$ and isolate $x^2$: $$x^2 = 4py$$

In the standard parabola equations, what does the parameter $p$ represent?

$p$ is the directed distance along the axis of symmetry from the vertex to the focus. $|p|$ is also the distance from the vertex to the directrix.

What is the standard form of a vertical parabola with vertex $(h, k)$?

$$(x - h)^2 = 4p(y - k)$$ If $p > 0$, it opens upward; if $p < 0$, it opens downward.

For the vertical parabola $(x - h)^2 = 4p(y - k)$, what are the coordinates of the focus and the equation of the directrix?

Focus: $$(h, k + p)$$ Directrix: $$y = k - p$$

What is the standard form of a horizontal parabola with vertex $(h, k)$?

$$(y - k)^2 = 4p(x - h)$$ If $p > 0$, it opens rightward; if $p < 0$, it opens leftward.

For the horizontal parabola $(y - k)^2 = 4p(x - h)$, what are the coordinates of the focus and the equation of the directrix?

Focus: $$(h + p, k)$$ Directrix: $$x = h - p$$

What is the latus rectum of a parabola, and what is its length?

The latus rectum is the chord passing through the focus perpendicular to the axis of symmetry, with endpoints lying on the parabola. Its length is $|4p|$.

What is the relationship between the leading coefficient $a$ in $y = a(x - h)^2 + k$ and the focal parameter $p$?

$$a = \frac{1}{4p} \quad \Longleftrightarrow \quad 4p = \frac{1}{a}$$

What is the axis of symmetry for the parabola $(x - h)^2 = 4p(y - k)$?

$$x = h$$

What is the axis of symmetry for the parabola $(y - k)^2 = 4p(x - h)$?

$$y = k$$

Find the vertex, focus, and directrix for the parabola $(x - 3)^2 = 8(y + 2)$.

Vertex: $(3, -2)$. Since $4p = 8$, $p = 2$. Focus: $(3, -2 + 2) = (3, 0)$. Directrix: $y = -2 - 2 \implies y = -4$.

Find the vertex, focus, and directrix for the parabola $(y + 1)^2 = -16(x - 5)$.

Vertex: $(5, -1)$. Since $4p = -16$, $p = -4$. Focus: $(5 + (-4), -1) = (1, -1)$. Directrix: $x = 5 - (-4) \implies x = 9$.

What is the equation of the parabola with focus $(0, -3)$ and directrix $y = 3$?

The vertex is midpoint of focus and directrix: $(0, 0)$. Focal parameter: $p = -3$. Standard vertical form: $$x^2 = 4(-3)y \implies x^2 = -12y$$

What is the equation of the parabola with vertex $(2, 1)$ and focus $(5, 1)$?

Horizontal axis with $p = 5 - 2 = 3$. Standard equation: $$(y - 1)^2 = 4(3)(x - 2)$$ $$(y - 1)^2 = 12(x - 2)$$

What reflective property makes parabolas essential in satellite dishes and headlights?

Any ray emitted from the focus reflects off the parabola parallel to the axis of symmetry; conversely, parallel incoming rays reflect to concentrate at the focus.

How do you find the endpoints of the latus rectum for $(x - h)^2 = 4p(y - k)$?

Start at the focus $(h, k + p)$ and move $|2p|$ units left and right: $(h - 2p, k + p) $ and $\quad (h + 2p, k + p)$

Parabola Equations, Focus, and Directrix

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