What Is a Parabola? Understanding Its Shape and Features

A parabola is a fundamental curve in coordinate geometry, classified as a conic section that arises when a plane cuts a right circular cone parallel to its slant edge. Parabolas are encountered extensively in mathematical problems, especially in the context of quadratic functions and locus-based geometry.

Formal Definition of Parabola via Locus Property

A parabola is the locus of all points in a plane that are equidistant from a fixed point, called the focus, and a fixed straight line, called the directrix.

Let the fixed point (focus) be $S(h, k+a)$ and the fixed line (directrix) be $y = k - a$. Let $P(x, y)$ be an arbitrary point on the locus. The distance from $P$ to the focus is $PS = \sqrt{(x - h)^2 + (y - (k + a))^2}$, and the perpendicular distance from $P$ to the directrix is $|y - (k - a)|$.

According to the locus definition, $PS =$ perpendicular distance from $P$ to directrix, i.e.:

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

Derivation of the Standard Equation of Parabola with Vertex at Origin

Consider the case where the vertex is at $(0,0)$, the focus at $(a,0)$, and the directrix as the line $x = -a$. Let $P(x, y)$ be a variable point on this locus. The focus is $S(a, 0)$. The distance from $P$ to the focus is $PS = \sqrt{(x - a)^2 + y^2}$. The perpendicular distance from $P$ to the directrix $x = -a$ is $|x + a|$. Hence, the locus condition gives:

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

Squaring both sides: $(x - a)^2 + y^2 = (x + a)^2$

Expanding both sides:

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

Subtracting $x^2$ and $a^2$ from both sides gives:

$-2ax + y^2 = 2ax$

Bringing like terms together:

$y^2 = 4ax$

Result: The equation of a parabola whose vertex is at the origin, axis is the $x$-axis, and focus at $(a, 0)$ is $y^2 = 4ax$.

A summary of various standard forms is discussed comprehensively under Parabola General Equations.

Standard Forms of Parabola Equations for Different Axes

The general equation $y^2 = 4ax$ refers to a parabola opening towards the positive $x$-axis. If the parabola opens towards the negative $x$-axis, the equation becomes $y^2 = -4ax$. For parabolas opening upwards or downwards, the axes change:

For a parabola opening upwards: $(x^2 = 4a y)$

For a parabola opening downwards: $(x^2 = -4a y)$

Here, $a$ is the distance from the vertex to the focus (also from the vertex to the directrix, in the opposite direction).

Geometric Elements: Vertex, Axis, Focus, Directrix, Latus Rectum

Vertex: The point about which the parabola is symmetric; for $y^2 = 4ax$, the vertex is at the origin $(0, 0)$.

Axis: The line passing through the focus and the vertex, which is the axis of symmetry. For $y^2 = 4ax$, the axis is the $x$-axis.

Focus: For $y^2 = 4ax$, the focus is at $S(a, 0)$.

Directrix: The directrix for $y^2 = 4ax$ is $x = -a$.

Latus Rectum: The chord through the focus perpendicular to the axis. Its length for $y^2 = 4ax$ is $4a$.

Detailed Deduction of the Parabola Latus Rectum Length

For the parabola $y^2 = 4ax$, the axis is the $x$-axis. The focus is at $(a,0)$. The equation of the line perpendicular to the axis passing through the focus is $x = a$.

Substitute $x = a$ into the parabola equation:

$y^2 = 4a(a) \implies y^2 = 4a^2$

Thus, $y = 2a$ and $y = -2a$. Therefore, the endpoints of the latus rectum are $(a, 2a)$ and $(a, -2a)$. The length is:

$\sqrt{(a - a)^2 + (2a - (-2a))^2} = \sqrt{0 + (4a)^2} = 4a$

Result: The length of the latus rectum for $y^2 = 4ax$ is $4a$.

Parametric Representation of Parabola Coordinates

Any point $P$ on the parabola $y^2 = 4ax$ can also be represented parametrically as $P(at^2, 2at)$, where $t$ is a real parameter. To verify, substitute $x = at^2$ and $y = 2at$: $(2at)^2 = 4a(at^2) \implies 4a^2 t^2 = 4a^2 t^2$, which is satisfied for all real $t$.

Parametric coordinates allow for convenient handling of tangents, normals, and chord problems on the parabola. 

Tangent to the Parabola at a Point (Derivation and Formula)

The equation to the tangent at a point $P(x_1, y_1)$ on the parabola $y^2=4ax$ is found as follows. The tangent at $P(x_1, y_1)$ is the line at $P$ with slope equal to the derivative at $P$. Start with $y^2 - 4ax = 0$.

Differentiate both sides with respect to $x$:

$2y \dfrac{dy}{dx} - 4a = 0$

$\dfrac{dy}{dx} = \dfrac{2a}{y}$

At point $P(x_1, y_1)$, the slope is $m = \dfrac{2a}{y_1}$. The equation of the tangent at $P$ is:

$y - y_1 = m(x - x_1)$

$y - y_1 = \dfrac{2a}{y_1}(x - x_1)$

Multiply both sides by $y_1$:

$y y_1 - y_1^2 = 2a (x - x_1)$

Bring all terms to one side:

$y y_1 = 2a (x + x_1)$

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

Equation of the Normal to a Parabola at a Point

The normal at $P(x_1, y_1)$ has the negative reciprocal of the tangent's slope. The slope is $-\dfrac{y_1}{2a}$. The equation is given by:

$y - y_1 = -\dfrac{y_1}{2a} (x - x_1)$

Multiply both sides by $2a$:

$2a(y - y_1) = -y_1(x - x_1)$ $2ay - 2a y_1 + y_1 x - y_1 x_1 = 0$

Rearrange:

$y_1 x + 2a y - (y_1 x_1 + 2a y_1) = 0$

Result: The equation of the normal to $y^2 = 4ax$ at $P(x_1, y_1)$ is $y_1 x + 2a y = y_1 x_1 + 2a y_1$.

Illustrative Example: Parabola Tangency

Given: Find the equation of the tangent to the parabola $y^2 = 8x$ at the point $(2, 4)$.

Substitution: Here, $a=2$, $x_1=2$, $y_1=4$. The tangent at $(x_1, y_1)$ has the equation $y y_1 = 2a(x + x_1)$. Thus, $y \cdot 4 = 4(x + 2)$.

Simplification: $4y = 4x + 8$

$y = x + 2$

Final result: The equation of the tangent is $y = x + 2$.

Intersection of a Line and a Parabola

Given the parabola $y^2 = 4ax$ and a line $y = mx + c$, substitute $y$ in the parabola equation:

$(mx + c)^2 = 4ax$

Expanding, $m^2 x^2 + 2 mc x + c^2 = 4a x$, bringing all terms to one side:

$m^2 x^2 + (2mc - 4a)x + c^2 = 0$

This quadratic equation in $x$ gives the points of intersection.

Important Locus and Lengths Associated with Parabola

If $P$ moves on the parabola $y^2 = 4ax$ and $Q$ is its projection on the directrix, then as $P(x, y)$, $Q(-a, y)$ and $PQ = |x + a|$. The focal length $SP = \sqrt{(x - a)^2 + y^2}$ as per initial definition, and for any point $(at^2, 2at)$, $SP = a(t^2 + 1)$.

The maximum and minimum value of a quadratic polynomial can be studied using the vertex form and properties of the parabola, elaborated in Max/Min Value of Quadratic Polynomial.

Pronunciation and Etymological Note

The word parabola is pronounced as /pəˈrabələ/. In various languages, parabolas are referenced in mathematical and literary contexts, e.g., parabola de los talentos, parabola del sembrador refer to parables, not to the geometric curve.

For mathematical comparison, distinctions between a parabola and other conics such as the hyperbola can be systematically explored using methods similar to those available under Trigonometry Basics.