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Hyperbola formula sheet

Every formula in the Hyperbola chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 10 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 equation and its parts

Standard hyperbola
8× asked, last 2026
x2a2−y2b2=1b2=a2(e2−1),c=ae=a2+b2\begin{aligned}&\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\\&b^2=a^2(e^2-1),\qquad c=ae=\sqrt{a^2+b^2}\end{aligned}
e>1e>1. Vertices (±a,0)(\pm a,0), foci (±ae,0)(\pm ae,0), directrices x=±a/ex=\pm a/e. c2=a2+b2c^2=a^2+b^2 (bb may exceed aa).
Latus rectum
19× asked, last 2026
LR=2b2a=2a(e2−1),ends (±ae, ±b2a)\text{LR}=\frac{2b^2}a=2a(e^2-1),\qquad \text{ends }\left(\pm ae,\ \pm\frac{b^2}a\right)
b2/ab^2/a is only half. Angle at the centre: tan⁡(θ/2)=(b2/a)/(ae)\tan(\theta/2)=(b^2/a)/(ae). Right angle at the other focus: b2/a=2aeb^2/a=2ae.
Transverse axis along y
5× asked, last 2026
−x2a2+y2b2=1:e2=1+a2b2foci (0,±be),LR=2a2b\begin{aligned}&-\frac{x^2}{a^2}+\frac{y^2}{b^2}=1:\quad e^2=1+\frac{a^2}{b^2}\\&\text{foci }(0,\pm be),\quad \text{LR}=\frac{2a^2}b\end{aligned}
Positive term holds the transverse axis. Conjugate hyperbola: 1/e2+1/e′2=11/e^2+1/e^{\prime2}=1.
Shifted hyperbola
4× asked, last 2026
(x−h)2a2−(y−k)2b2=1,foci (h±ae, k)\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1,\qquad \text{foci }(h\pm ae,\ k)
Centre = midpoint of foci; 2ae2ae = their distance. A curve point on the focal line is a vertex.

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6 more formulas in 2 topics

Focus, directrix and focal distances; and Rectangular hyperbola, ellipses and lines. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Focus, directrix and focal distances

  • Focus–directrix5×
  • Focal distances6×
  • Fixing a, b, e12×

Rectangular hyperbola, ellipses and lines

  • Rectangular hyperbola3×
  • Ellipse–hyperbola links12×
  • Line meets hyperbola4×

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About 65 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

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