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Ellipse formula sheet

Every formula in the Ellipse 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 ellipse

Major axis along x
6× asked, last 2023
x2a2+y2b2=1 (a>b>0),b2=a2(1−e2)foci (±ae,0),directrices x=±ae\begin{aligned}&\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\ (a>b>0),\qquad b^2=a^2(1-e^2)\\&\text{foci }(\pm ae,0),\qquad \text{directrices }x=\pm\frac ae\end{aligned}
0<e<10<e<1; e2=1−b2/a2e^2=1-b^2/a^2 (not b/ab/a). Focus to its directrix: a(1−e2)/ea(1-e^2)/e.
Major axis along y
9× asked, last 2026
b>a:a2=b2(1−e2),foci (0,±be),ℓ=2a2bb>a:\quad a^2=b^2(1-e^2),\quad \text{foci }(0,\pm be),\quad \ell=\frac{2a^2}b
The larger denominator is (semi-major)² and sets ee, foci, directrices y=±b/ey=\pm b/e.
Latus rectum
18× asked, last 2026
ℓ=2b2a=2a(1−e2),ends (±ae, ±b2a)\ell=\frac{2b^2}a=2a(1-e^2),\qquad \text{ends }\left(\pm ae,\ \pm\frac{b^2}a\right)
Denominator = semi-major axis. e=2/3e=2/3, ℓ=10\ell=10: a=9a=9. e=1/2e=1/\sqrt2: ℓ=a\ell=a.
Finding e
8× asked, last 2026
b=k ae⇒k2e2=1−e2,e2=11+k2b=k\,ae\Rightarrow k^2e^2=1-e^2,\qquad e^2=\frac1{1+k^2}
Write lengths in aa, ee; aa cancels. Minor axis 2b2b, foci 2ae2ae apart.

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12 more formulas in 4 topics

Directrix and shifted centre; Focal distances, parametric form and area; Lines, chords and circles; and Areas, loci and other conics. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Directrix and shifted centre

  • Focus–directrix4×
  • Shifted ellipse5×

Focal distances, parametric form and area

  • Focal distances8×
  • Focal chord
  • Parametric point4×

Lines, chords and circles

  • Line meets ellipse8×
  • Chord with given midpoint6×
  • Tangent3×
  • Circle and ellipse7×

Areas, loci and other conics

  • Figures6×
  • Locus5×
  • With hyperbola or parabola13×

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