# Hyperbola Formulas

Want to solve all Hyperbola equations in an easy way? Then, this page helps you a lot. Because we have curated simple hyperbola formulas to simplify the problems quickly and effortlessly. So, try to remember the hyperbola formulae for all concepts by using the Hyperbola Formula Cheat Sheet existed here. Also, you can make your calculations so easy at any time if you memorize the formulas of hyperbola.

## Hyperbola Formulas List | Cheatsheet of Hyperbola Formulae

Make the most out of these Hyperbola Formulas and solve the calculations within given time. The list of Hyperbola formulae that exist here helps you to do your homework or math assignments at a faster pace.

**1. Standard equation of Hyperbola**

\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1

- Length of transverse axis → 2a
- Length of conjugate axis → 2b
- Directrix: x = a/e and x = – a/e
- Focus: S (ae, 0) and S’ (- ae, 0)
- Length of Latus Rectum is given by \(\frac{2 b^{2}}{a}\)

**2. Eccentricity**

(A) For the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1, b^{2} = a^{2} (e^{2} – 1)

(B) Equation of vertical hyperbola is \(\frac{y^{2}}{b^{2}}-\frac{x^{2}}{a^{2}}\) = 1

Length of L.R. = \(\frac{2 a^{2}}{b}\) also a^{2} = b^{2} (e^{2} – 1)

3. The equation ax^{2} + by^{2} + 2hxy + 2gx + 2fy + c = 0 will represent an hyperbola if h^{2} – ab > 0 & Δ = abc + 2fgh – af^{2} – bg^{2} – ch^{2} ≠ 0.

**4. Conjugate Hyperbola**

(i) The equation of the conjugate hyperbola of \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 is – \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\) = 1

(ii) If e_{1} and e_{2} are the eccentricities of the hyperbola and its conjugate then

\(\frac{1}{e_{1}^{2}}+\frac{1}{e_{21}^{2}}\)

**5. The equation of hyperbola in the parametric form will be given by x = a sec Φ, y = b tan Φ**

**6. Condition of tangency**

The line y = mx + c touches the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1, if c = ± \(\sqrt{a^{2} m^{2}-b^{2}}\) and point of tangency is \(\left(-\frac{\mathrm{a}^{2} \mathrm{m}}{\mathrm{c}},-\frac{\mathrm{b}^{2}}{\mathrm{c}}\right)\)

**7. Equation of tangent**

(i) The equation of the tangent at any point (x_{1}, y_{1}) on the hyperbola

\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \text { is } \frac{x x_{1}}{a^{2}}-\frac{y y_{1}}{b^{2}}=1\)

(ii) Equation of tangent to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 at the point (a sec θ, b tan θ) is \(\frac{x}{a}\) sec θ – \(\frac{y}{b}\) tan θ = 1.

(iii) Slope form: y = mx ± \(\sqrt{a^{2} m^{2}-b^{2}}\) and the point of contacts is \(\left(\pm \frac{\mathrm{a}^{2} \mathrm{m}}{\sqrt{\mathrm{a}^{2} \mathrm{m}^{2}-\mathrm{b}^{2}}}, \pm \frac{\mathrm{b}^{2}}{\sqrt{\mathrm{a}^{2} \mathrm{m}^{2}-\mathrm{b}^{2}}}\right)\)

**8. Equation of the normal**

(i) The equation of normal to tbe hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 at (x_{1}, y_{1}) is

\(\frac{a^{2} x}{x_{1}}+\frac{b^{2} y}{y_{1}}\) = a^{2} + b^{2} = a^{2}e^{2}.

(ii) The equation of normal at (a sec θ, b tan θ) to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 is ax cos θ + by cot θ = a^{2} + b^{2}.

(iii) Slope form: y = mx – \(\frac{m\left(a^{2}+b^{2}\right)}{\sqrt{a^{2}-b^{2} m^{2}}}\)

**9. Pair of Tangents SS _{1} = T^{2}**

**10. Chord of Contact T = 0 at (x _{1}, y_{1})**

**11. Equation of the chord whose middle point is given T = S _{1}**

**12. Director Circle**

The equation of the director circle is x^{2} + y^{2} = a^{2} – b^{2}

**13. Diameter**

If y = mx + c represent a system of parallel chords of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) then the equation of the diameter is y = \(\frac{b^{2}}{a^{2} m}\)x.

**14. Asymptotes**

Equation of the asymptotes of hyperbolas \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 and –\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\) = 1 are y = ±\(\frac{b}{a}\)x.

**15. Equation of rectangular hyperbola**

Hyperbola whose eccentricity is \(\sqrt{2}\), equation is x^{2} – y^{2} = a^{2}. Equation of hyperbola referred asymptotes as axes is xy = c^{2} where c^{2} = \(\frac{a^{2}+b^{2}}{4}\). Point on xy = c^{2} may be taken (ct, c/t)

16. Equation of chord joining points t_{1} and t_{2} on the hyperbola xy = c^{2} is x + yt_{1}t_{2} – c(t_{1} + t_{2}) = 0

17. Tangent at the point “t” to the:

x + yt^{2} – 2ct = 0

If we call the point t i.e. (ct, c/t) as (x_{1}, y_{1}) then above tangent can be written as \(\frac{x}{x_{1}}+\frac{y}{y_{1}}=2\)

18. Normal to the Hyperbola at point “t”

xt^{3} – yt – ct^{4} + c = 0

in another form as (ct, c/t) = (x_{1}, y_{1})

xx_{1} – yy_{1} = x_{1}^{2} – y_{1}^{2}

**19. Equation of Auxiliary circle**

if hyperbola is \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 is x^{2} + y^{2} = a^{2}

20. If e_{1} and e_{2} be the eccentricities of a hyperbola and its conjugate then \(\frac{1}{e_{1}^{2}}+\frac{1}{e_{2}^{2}}=1\)