Hyperbola Formulas

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, b² = a² (e² – 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 a² = b² (e² – 1)

3. The equation ax² + by² + 2hxy + 2gx + 2fy + c = 0 will represent an hyperbola if h² – ab > 0 & Δ = abc + 2fgh – af² – bg² – ch² ≠ 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 e₁ and e₂ 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 (x₁, y₁) 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 (x₁, y₁) is
\(\frac{a^{2} x}{x_{1}}+\frac{b^{2} y}{y_{1}}\) = a² + b² = a²e².

(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 = a² + b².

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

9. Pair of Tangents SS₁ = T²

10. Chord of Contact T = 0 at (x₁, y₁)

11. Equation of the chord whose middle point is given T = S₁

12. Director Circle

The equation of the director circle is x² + y² = a² – b²

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 x² – y² = a². Equation of hyperbola referred asymptotes as axes is xy = c² where c² = \(\frac{a^{2}+b^{2}}{4}\). Point on xy = c² may be taken (ct, c/t)

16. Equation of chord joining points t₁ and t₂ on the hyperbola xy = c² is x + yt₁t₂ – c(t₁ + t₂) = 0

17. Tangent at the point “t” to the:

x + yt² – 2ct = 0
If we call the point t i.e. (ct, c/t) as (x₁, y₁) then above tangent can be written as \(\frac{x}{x_{1}}+\frac{y}{y_{1}}=2\)

18. Normal to the Hyperbola at point “t”
xt³ – yt – ct⁴ + c = 0
in another form as (ct, c/t) = (x₁, y₁)
xx₁ – yy₁ = x₁² – y₁²

19. Equation of Auxiliary circle

if hyperbola is \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\) = 1 is x² + y² = a²

20. If e₁ and e₂ be the eccentricities of a hyperbola and its conjugate then \(\frac{1}{e_{1}^{2}}+\frac{1}{e_{2}^{2}}=1\)

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