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Parabola revision notes

Every idea in the Parabola chapter for JEE Main, with the usual mistakes and the fast methods. 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 concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Standard forms and the focus–directrix definition

Four standard parabolas
3× asked, last 2023
y2=±4ax: S(±a,0), x=∓a,x2=±4ay: S(0,±a), y=∓ay^2=\pm4ax:\ S(\pm a,0),\ x=\mp a,\qquad x^2=\pm4ay:\ S(0,\pm a),\ y=\mp a
a>0a>0, vertex (0,0)(0,0). The curve opens towards the focus; the directrix is on the other side of the vertex.
y2=4axy^2=4axy2=−4axy^2=-4axx2=4ayx^2=4ayx2=−4ayx^2=-4ay
Each opens towards its focus (red dot); the directrix (dashed) is on the other side of the vertex
Latus rectum and focal length
4× asked, last 2022
LR=4a,ends (a,±2a),VS=a,S to directrix=2a\text{LR}=4a,\qquad \text{ends }(a,\pm2a),\qquad VS=a,\quad S\text{ to directrix}=2a
For y2=4axy^2=4ax. aa is the vertex–focus gap wherever the vertex sits. LR is the chord through SS perpendicular to the axis.
xxyyS(a,0)S(a,0)(a,2a)(a,2a)(a,−2a)(a,-2a)PPMMx=−ax=-a
LR =4a=4a through SS; every point has SP=PMSP=PM (distance to the directrix)
  • Not 2a2a. That is the semi-latus rectum (focus to one end). The latus rectum is 4a4a, the whole coefficient of xx: y2=6kxy^2=6kx has LR 6k6k, not 3k3k.
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, then read 4a4a. Axis y=ky=k. Same shift for (x−h)2=4a(y−k)(x-h)^2=4a(y-k): S(h,k+a)S(h,k+a), y=k−ay=k-a.
xxyyV(a2,0)V(\frac a2,0)S(a,0)S(a,0)directrix x=0x=0
y2=2a(x−a2)y^2=2a(x-\frac a2) (a=1a=1 here): the vertex is (a2,0)(\frac a2,0) and 4A=2a4A=2a, so the directrix is x=0x=0, not −a2-\frac a2
  • y=px2+bx+cy=px^2+bx+c: vertex at x=−b/2px=-b/2p, vertex height c−b2/4pc-b^2/4p, and 4a=1/∣p∣4a=1/|p|. It opens upward when p>0p>0.
  • Not measured from the old origin. Focus and directrix move with the vertex. y2=2a(x−a/2)y^2=2a(x-a/2) has vertex (a/2,0)(a/2,0), so its directrix is x=a/2−a/2=0x=a/2-a/2=0, not x=−a/2x=-a/2.
Parabola 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}
Focus (α,β)(\alpha,\beta), directrix lx+my+n=0lx+my+n=0, any tilt. The vertex is midway from SS to the directrix along the axis.
xxyyV(32,3)V(\frac32,3)S=2V−F=(3,6)S=2V-F=(3,6)FFx+2y=0x+2y=0
A tilted parabola: every point is as far from SS as from the directrix; VV is midway from FF to SS
  • Quick method. Given the vertex VV and the directrix: drop the foot FF from VV to the directrix, then S=2V−FS=2V-F. V(3/2,3)V(3/2,3), directrix x+2y=0x+2y=0: F=(0,0)F=(0,0), so S=(3,6)S=(3,6).

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12 more concepts in 5 topics

Parametric form and focal distance; Chords and focal chords; Lines and chords with a given midpoint; and Areas, loci, distances and images, plus ideas from circles, quadratics, differential equations. 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 joining t1 and t27×
  • Focal chord13×
  • Focal segments (harmonic mean)4×

Lines and chords with a given midpoint

  • Line meets a parabola9×
  • Chord with a given midpoint3×

Areas, loci, distances and images

  • Triangle with parametric vertices5×
  • Equilateral triangle at the vertex3×
  • Locus with a parameter8×
  • Shortest distance from a point7×
  • Mirror image of a parabola3×

Also asked here: circles, quadratics, differential equations

  • Key points4

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