Understanding the General Equation of a Parabola

A parabola is a plane curve defined as the set of all points equidistant from a fixed point (called the focus) and a fixed straight line (called the directrix). The general equations of parabolas are central to coordinate geometry and conic sections.

Formal Mathematical Structure and General Equations of a Parabola

The general equation of a parabola with its axis parallel to the coordinate axes and vertex at $(h, k)$ takes one of two forms.

If the axis is parallel to the $x$-axis, the equation is given by

$y^2 = 4a(x - h) + 2k y + k^2$.

More commonly, the equation is simplified as

$(y - k)^2 = 4a(x - h)$.

If the axis is parallel to the $y$-axis, the general equation is

$(x - h)^2 = 4a(y - k)$.

Definition of the Parabola as a Locus

A parabola is defined as the locus of all points in a plane that are equidistant from a fixed point (focus) and a fixed straight line (directrix).

Let the focus be $F(a, 0)$ and the directrix be the line $x = -a$. For any point $P(x, y)$ on the parabola, the distance $PF$ to the focus is equal to the perpendicular distance $PD$ from $P$ to the directrix.

Derivation of the Standard Parabola Equation Using the Locus Property

Let $PF = PD$. The coordinates of the focus are $F(a, 0)$. For point $P(x, y)$, the distance to the focus is

$PF = \sqrt{(x - a)^2 + (y - 0)^2}$

$PF = \sqrt{(x - a)^2 + y^2}$

The equation for the directrix is $x = -a$. The perpendicular distance from $P(x, y)$ to the directrix is

$PD = |x + a|$

By the definition of parabola, set the distances equal:

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

Square both sides:

$(x - a)^2 + y^2 = (x + a)^2$

Expand each side:

$x^2 - 2a x + a^2 + y^2 = x^2 + 2a x + a^2$

Subtract $x^2 + a^2$ from both sides:

$-2a x + y^2 = 2a x$

Bring like terms together:

$y^2 = 2a x + 2a x$

$y^2 = 4a x$

This is the standard equation of a parabola with vertex at $(0, 0)$, axis along the $x$-axis and opening towards the positive $x$-direction.

Geometric Elements Determined by the Parabola Equation

For the general equation $(y - k)^2 = 4a(x - h)$, the following geometric features are determined:

The vertex is at $(h, k)$.

The axis of symmetry is the line $y = k$.

The focus is $(h + a, k)$.

The directrix is $x = h - a$.

The length of the latus rectum is $4a$.

Variants of the Standard Parabola Equation: All Orientations

The four canonical orientations of the parabola, with their axes aligned with coordinate axes, are as follows:

$y^2 = 4a x$     (Opens right, axis along $x$)

$y^2 = -4a x$     (Opens left, axis along $x$)

$x^2 = 4a y$     (Opens up, axis along $y$)

$x^2 = -4a y$     (Opens down, axis along $y$)

Parametric Representation of Points on a Parabola

For the parabola $y^2 = 4a x$, all points on the curve can be represented parametrically as $(a t^2, 2a t)$, where $t \in \mathbb{R}$.

General Quadratic Form and Reduction to Parabola Standard Form

The most general form of a parabolic equation in the plane is $A x^2 + B x y + C y^2 + D x + E y + F = 0$, where $B^2 - 4AC = 0$ indicates the equation represents a parabola.

To reduce the general quadratic $y = a x^2 + b x + c$ to its standard form, complete the square in $x$:

Start with $y = a x^2 + b x + c$.

Extract $a$, $y = a \left( x^2 + \frac{b}{a} x \right) + c$.

Complete the square in $x$:

$x^2 + \frac{b}{a} x = \left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2}$

Substitute this expression:

$y = a \left[ \left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2} \right] + c$

$y = a \left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c$

$y = a \left(x + \frac{b}{2a}\right)^2 + \left( c - \frac{b^2}{4a} \right )$

Thus, in standard vertex form: $y = a (x - h)^2 + k$ with $h = -\frac{b}{2a}$ and $k = c - \frac{b^2}{4a}$.

Parabola General Equations

Important Chords and Tangents on a Parabola

The equation of the tangent to $y^2 = 4a x$ at point $(x_1, y_1)$ is $y y_1 = 2a (x + x_1)$.

The equation of the normal to $y^2 = 4a x$ at $(x_1, y_1)$ is $y - y_1 = -\frac{y_1}{2a}(x - x_1)$.

r

A chord joining points $(a t_1^2, 2a t_1)$ and $(a t_2^2, 2a t_2)$ has equation $y - 2a t_1 = \frac{2}{t_1 + t_2} (x - a t_1^2)$.

Illustrative Example of Converting a General Parabolic Equation

Example

Given: $y = 2 x^2 - 4 x + 3$. Find the vertex, focus, directrix, and length of latus rectum.

Rewrite in completed square form:
$y = 2 (x^2 - 2x) + 3$

$x^2 - 2x = (x - 1)^2 - 1$

$y = 2 (x - 1)^2 - 2 + 3$

$y = 2 (x - 1)^2 + 1$

This matches $y = a (x - h)^2 + k$ with $a = 2$, $h = 1$, $k = 1$.

To express in the form $(y - 1) = 2 (x - 1)^2$ or, equivalently, $(y - 1) = 4a' (x - 1)^2$ with $4a' = 2$, $a' = \frac{1}{2}$.

The vertex is $(1, 1)$.

The axis of symmetry is $x = 1$.

Since the parabola opens upward, the focus is at $(1, 1 + \frac{1}{4a'}) = \left(1, 1 + \frac{1}{2}\right) = (1, \frac{3}{2})$.

The directrix is $y = 1 - \frac{1}{4a'} = 1 - \frac{1}{2} = \frac{1}{2}$.

The length of the latus rectum is $\frac{1}{|a|} = \frac{1}{2}$.

Eccentricity and Distinguishing Features of Parabolas

For all parabolas, the eccentricity $e = 1$.

Exam Cues Related to Parabola General Equations

Exam Tip: To determine if a quadratic equation represents a parabola, ensure the discriminant $B^2 - 4AC = 0$ in the general conic $A x^2 + B x y + C y^2 + D x + E y + F = 0$.

When converting from general to standard form, always perform completing the square separately for $x$ and $y$ terms depending on which variable is squared, as exam errors often arise from omitting the constant shift or sign errors