8. ANALYTICAL GEOMETRY: PARABOLA

Introduction

A parabola is a curve formed by all points in a plane that are at equal distance from a fixed point called the focus and a fixed straight line called the directrix. It is one of the most important curves in coordinate geometry and appears in physics, engineering, and optics.

Standard Definition of a Parabola

A parabola is the set of all points P(x, y) such that:

Distance (P to Focus) = Distance (P to Directrix)

Standard Forms of Parabola

1. Parabola Opening Right

y² = 4ax

Focus: (a, 0)
Directrix: x = −a
Vertex: (0, 0)
Axis: X-axis
Latus Rectum: line x = a
Length of Latus Rectum = 4a

2. Parabola Opening Left

y² = −4ax

Focus: (−a, 0)
Directrix: x = a

3. Parabola Opening Upward

x² = 4ay

Focus: (0, a)
Directrix: y = −a
Length of Latus Rectum = 4a

4. Parabola Opening Downward

x² = −4ay

Focus: (0, −a)
Directrix: y = a

Geometrical Features of a Parabola

Vertex

The turning point of the parabola. For standard forms, vertex is at (0, 0).

Focus

The fixed point used in the definition of parabola.

Directrix

The fixed line used in the parabola definition.

Axis

A line passing through the vertex and focus.

Latus Rectum

A line segment perpendicular to the axis, passing through the focus.

Length of latus rectum = 4a

Parametric Coordinate of a Point on a Parabola

For parabola y² = 4ax, coordinates of any point on the curve can be written as:

(at², 2at)

Parameter t makes calculations easier in tangent and normal problems.

Equation of Tangent to a Parabola

1. Tangent at Point (x₁, y₁)

For parabola y² = 4ax:

yy₁ = 2a(x + x₁)

2. Tangent in Slope Form

y = mx + a/m

3. Tangent at Parametric Point (at², 2at)

ty = x + at²

Equation of Normal to a Parabola

Normal at Parametric Point (at², 2at)

y − 2at = −t(x − at²)

Normal in Slope Form

If slope of tangent = m, slope of normal = −1/m.

Chord of Contact

If a line touches the parabola at points P and Q, then from an external point (x₁, y₁), the chord of contact is:

yy₁ = 2a(x + x₁)

Director Circle

For a standard parabola, the set of points from which two tangents drawn are at right angles form a locus known as the director circle.

For y² = 4ax, the director circle is:

x² + y² = 2a²

Focal Chord

A chord passing through the focus.

If the endpoints of focal chord have parameters t₁ and t₂:

t₁t₂ = −1

Length of Subtangent and Subnormal

Subtangent

For y² = 4ax at parametric point (at², 2at):

Subtangent = y₁ / m

Subnormal

Subnormal = y₁ × m

(m is slope of tangent)

Important Results

Focus of y² = 4ax is (a, 0)
Length of latus rectum = 4a
Parametric form = (at², 2at)
Tangent: yy₁ = 2a(x + x₁)
Normal: y − 2at = −t(x − at²)

Examples (Conceptual Overview)

Finding equation of tangent at a point
Finding normal using parameter t
Converting general equation to standard form
Finding length of latus rectum
Solving locus questions