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Parabola formula sheet

Every formula in the Parabola chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 16 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that formula, and "last" is the latest year one appeared. ★ marks formulas asked 8 or more times.

Standard forms and focus–directrix

Four standard parabolas
3× asked, last 2023
y2=±4ax: S(±a,0), x=∓ax2=±4ay: S(0,±a), y=∓a\begin{aligned}&y^2=\pm4ax:\ S(\pm a,0),\ x=\mp a\\&x^2=\pm4ay:\ S(0,\pm a),\ y=\mp a\end{aligned}
a>0a>0, vertex O; opens towards the focus.
Latus rectum
4× asked, last 2022
LR=4a,ends (a,±2a)VS=a,S to directrix=2a\begin{aligned}&\text{LR}=4a,\qquad \text{ends }(a,\pm2a)\\&VS=a,\quad S\text{ to directrix}=2a\end{aligned}
2a2a is the semi-latus rectum. y2=6kxy^2=6kx: LR 6k6k.
Shifted parabola
8× asked, last 2026
(y−k)2=4a(x−h):V(h,k), S(h+a,k), x=h−a(y-k)^2=4a(x-h):\quad V(h,k),\ S(h+a,k),\ x=h-a
Complete the square first. y=px2+bx+cy=px^2+bx+c: vertex x=−b/2px=-b/2p, 4a=1/∣p∣4a=1/|p|.
From focus and directrix
8× asked, last 2025
SP=PM:(x−α)2+(y−β)2=(lx+my+n)2l2+m2SP=PM:\quad (x-\alpha)^2+(y-\beta)^2=\frac{(lx+my+n)^2}{l^2+m^2}
Vertex midway from SS to the directrix. Given vertex VV and directrix foot FF: S=2V−FS=2V-F.

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12 more formulas in 4 topics

Parametric form and focal distance; Chords and focal chords; Lines and chords; and Areas, loci, distances and images. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Parametric form and focal distance

  • Parametric point7×
  • Focal distance5×

Chords and focal chords

  • Chord t₁t₂7×
  • Focal chord13×
  • Focal segments4×

Lines and chords

  • Line meets parabola9×
  • Chord with given midpoint3×

Areas, loci, distances and images

  • Triangle area5×
  • Equilateral at the vertex3×
  • Locus with a parameter8×
  • Shortest distance7×
  • Mirror image3×

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