Parabola

General Equation of a Conic (Focal Directrix Property)

The general equation of a conic with focus (p, q) and directrix lx + my + n = 0 is:

(l² + m²)[(x – p)² + (y – q)²] = e²(lx + my + n)²

Which can be written as: ax² + 2hxy + by² + 2gx + 2fy + c = 0

Case (i): Focus lies on the directrix

• D = abc + 2fgh – af² – bg² – ch² = 0
• If e > 1, h² > ab → real & distinct lines
• If e = 1, h² = ab → coincident lines
• If e < 1, h² < ab → imaginary lines

Case (ii): Focus does not lie on the directrix

Conic	Condition
Parabola	e = 1, D ≠ 0, h² = ab
Ellipse	0 < e < 1, D ≠ 0, h² < ab
Hyperbola	e > 1, D ≠ 0, h² > ab
Rectangular Hyperbola	e > 1, D ≠ 0, h² > ab, a + b = 0

Standard Parabola y² = 4ax

• Vertex: (0, 0)
• Focus: (a, 0)
• Axis: y = 0
• Directrix: x + a = 0

Latus Rectum

• Length = 4a
• Semi-latus rectum = 2a
• Ends: (a, 2a) and (a, –2a)

Parametric Representation

Coordinates of a point on parabola y² = 4ax: (at², 2at)

Types of Parabola

• y² = 4ax
• y² = –4ax
• x² = 4ay
• x² = –4ay

Position of a Point

For parabola y² = 4ax, point (x₁, y₁):

• Outside if y₁² – 4ax₁ > 0
• On if y₁² – 4ax₁ = 0
• Inside if y₁² – 4ax₁ < 0

Chord Joining Two Points

Equation of chord joining P(t₁) and Q(t₂): y(t₁ + t₂) = 2x + 2at₁t₂

• If PQ is the focal chord → t₁t₂ = –1
• If t₁t₂ = k → chord passes through (–ka, 0)

Tangent to Parabola y² = 4ax

• Point form: yy₁ = 2a(x + x₁)
• Slope form: y = mx + a/m
• Parametric form: ty = x + at²

Normal to Parabola y² = 4ax

• Point form: y – y₁ = –(x – x₁)(y₁ / 2a)
• Slope form: y = mx – 2am – am³
• Parametric form: y + tx = 2at + at³

Chord of Contact

Equation: yy₁ = 2a(x + x₁)

Chord with Given Middle Point

Equation: y – y₁ = (x – x₁)(y₁ / 2a)

Conormal Points

• Sum of slopes of three concurrent normals = 0
• Sum of ordinates of three conormal points = 0
• Centroid lies on the axis of the parabola

Important Highlights

• Tangent and normal at point P bisect the angle between the focal radius and the perpendicular to the directrix.
• The tangent cut off between the directrix and curve subtends a right angle at the focus.
• Tangents at extremities of the focal chord intersect at right angles on the directrix.
• Any tangent and perpendicular from the focus meet at the vertex.
• Semi-latus rectum is the harmonic mean of the segments of the focal chord.