Parabola

Definition

• A parabola is the locus of a point whose distance from a fixed point (focus) is equal to its distance from a fixed straight line (directrix).
• Eccentricity of a parabola: e = 1

Basic Terms

• Axis: Line through focus perpendicular to directrix
• Vertex: Midpoint of focus and directrix
• Latus rectum: Chord through focus perpendicular to axis
• Length of latus rectum: 4a
• Focal chord: Any chord passing through focus
• Double ordinate: Any chord perpendicular to axis

Standard Forms of Parabola

y² = 4ax (opens right)

• Vertex: (0, 0)
• Focus: (a, 0)
• Directrix: x = −a
• Axis: y = 0
• Latus rectum: x = a
• Ends of latus rectum: (a, ±2a)

y² = −4ax (opens left)

• Focus: (−a, 0)
• Directrix: x = a

x² = 4ay (opens upward)

• Focus: (0, a)
• Directrix: y = −a

x² = −4ay (opens downward)

• Focus: (0, −a)
• Directrix: y = a

Parametric Coordinates

• For y² = 4ax:
• x = at², y = 2at
• Point corresponding to parameter t is called point t on the parabola

Equation of a Chord

• Chord joining points (x₁, y₁) and (x₂, y₂) on y² = 4ax:
• y(y₁ + y₂) = 4ax + y₁y₂
• Chord joining parametric points t₁ and t₂:
• y(t₁ + t₂) = 2(x + at₁t₂)

Focal Chord

• Condition for chord joining t₁, t₂ to be focal chord:
• t₁t₂ = −1
• Length of focal chord:
• a(t₂ − t₁)²
• Length of focal chord through point t:
• a(t + 1/t)²

Equation of Tangent

Point Form

• Tangent at point (x₁, y₁) on y² = 4ax:
• yy₁ = 2a(x + x₁)

Parametric Form

• Tangent at parametric point t:
• ty = x + at²

Slope Form

• y = mx + a/m
• Point of contact: (a/m², 2a/m)

Condition for Tangency

• Line y = mx + c touches y² = 4ax if:
• c = a/m

Point of Intersection of Tangents

• Tangents at points t₁ and t₂ intersect at:
• (at₁t₂, a(t₁ + t₂))
• Angle between tangents:
• tanθ = |(t₂ − t₁)/(1 + t₁t₂)|

Equation of Normal

Point Form

• y − y₁ = −(x − x₁)y₁/(2a)

Parametric Form

• y + tx = 2at + at³

Slope Form

• y = mx − 2am − am³

Position of Point & Line

• Point (x₁, y₁) lies:
• Outside if y₁² − 4ax₁ > 0
• On if y₁² − 4ax₁ = 0
• Inside if y₁² − 4ax₁ < 0
• Line y = mx + c cuts parabola in:
• Two / one / no real points if a − mc > = < 0

Pair of Tangents

• From point (x₁, y₁) to y² = 4ax:
• SS₁ = T²
• Where:
• S = y² − 4ax
• S₁ = y₁² − 4ax₁
• T = yy₁ − 2a(x + x₁)

Chord of Contact

• From point (x₁, y₁):
• yy₁ − 2a(x + x₁) = 0
• Chord of contact of perpendicular tangents always passes through focus

Pole & Polar

• Polar of point (x₁, y₁):
• yy₁ − 2a(x + x₁) = 0
• Pole of line lx + my + n = 0:
• (n/l, 2am/l)
• Polar of focus is directrix and vice versa

Diameter of a Parabola

• Diameter is locus of midpoints of parallel chords
• Diameter bisecting chords of slope m:
• y = 2a/m