FORMULA LIBRARY : PARABOLA

PARABOLA

— JEE MATH APEX —

JEE MATH APEX

1. Standard Forms of Parabola

• Right-opening: y² = 4ax (focus: (a,0), vertex: (0,0), axis: y=0)
• Left-opening: y² = −4ax (focus: (−a,0))
• Upward-opening: x² = 4ay (focus: (0,a))
• Downward-opening: x² = −4ay (focus: (0,−a))
• Latus rectum length: 4a
• Directrix: x = −a (for y²=4ax), y = −a (for x²=4ay)

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2. Parametric Form

• For y² = 4ax: x = at², y = 2at
• Point: (at², 2at) — parameter t
• For x² = 4ay: x = 2at, y = at²
• Point: (2at, at²)
• Focus in parametric: t = 1 for y²=4ax

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3. Tangent to Parabola

• At point (x₁,y₁) on y²=4ax: yy₁ = 2a(x + x₁)
• At parametric point (at², 2at): ty = x + at²
• With slope m: y = mx + a/m
• Point of contact: (a/m², 2a/m)
• Condition for y = mx + c to be tangent: c = a/m

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4. Normal to Parabola

• At point (x₁,y₁) on y²=4ax: y − y₁ = (−y₁/2a)(x − x₁)
• At parametric point (at², 2at): y = −tx + 2at + at³
• With slope m: y = mx − 2am − am³
• Foot of normal: (am², −2am)
• Three normals can be drawn from a point to parabola

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5. Chord of Parabola

• Chord joining t₁ and t₂: y(t₁+t₂) = 2x + 2at₁t₂
• Focal chord (passes through focus): t₁t₂ = −1
• Length of focal chord with parameter t: a(t + 1/t)²
• If t₁, t₂ are endpoints of focal chord: t₁t₂ = −1

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6. Chord of Contact

• Chord of contact from (x₁,y₁) to y²=4ax: yy₁ = 2a(x + x₁)
• Pair of tangents from (x₁,y₁): SS₁ = T²
• Where S = y² − 4ax, S₁ = y₁² − 4ax₁, T = yy₁ − 2a(x+x₁)
• Chord of contact is perpendicular to line joining focus and point

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7. Chord with Midpoint

• Chord with midpoint (x₁,y₁) of y²=4ax: T = S₁
• yy₁ − 2a(x+x₁) = y₁² − 4ax₁
• Slope of chord: 2a/y₁

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8. Director Circle

• Director circle of y²=4ax is the directrix: x = −a
• Locus of intersection of perpendicular tangents is the directrix
• Perpendicular tangents to parabola intersect on directrix

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9. Properties of Focal Chord

• If focal chord makes angle θ with axis: Length = 4a cosec²θ
• Sum of reciprocals of focal distances: 1/SP + 1/SQ = 1/a (constant)
• Product of focal distances: SP × SQ = a × (length of focal chord)
• Focal chord is minimum at latus rectum

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10. Position of a Point

• For parabola y² = 4ax and point (x₁,y₁):
• If y₁² − 4ax₁ < 0: Point inside
• If y₁² − 4ax₁ = 0: Point on parabola
• If y₁² − 4ax₁ > 0: Point outside

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11. Reflection Property

• Tangent at any point makes equal angles with the focal radius and the line parallel to axis
• Angle of incidence = Angle of reflection
• Rays parallel to axis reflect through focus
• Tangent bisects the angle between focal radius and perpendicular to directrix

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12. General Equation

• General equation: (ax + by)² + 2gx + 2fy + c = 0
• If axis is parallel to x-axis: x = ay² + by + c
• If axis is parallel to y-axis: y = ax² + bx + c
• Vertex form: (y − k)² = 4a(x − h) (vertex at (h,k))

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13. Two Tangents from External Point

• Two tangents from external point to y²=4ax are real if point is outside
• Angle between tangents: tan θ = |2√(y₁²−4ax₁)/(x₁+a)|
• Tangents are perpendicular if point lies on directrix (x = −a)
• Length of tangent from external point: √(S₁)

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14. Important Theorems

• If normal at t₁ meets parabola again at t₂: t₂ = −t₁ − 2/t₁
• If normals at t₁ and t₂ meet on parabola: t₁t₂ = 2
• Tangents at extremities of focal chord intersect at right angle on directrix
• Area of triangle formed by three points on parabola = (a²/2)|(t₁−t₂)(t₂−t₃)(t₃−t₁)|