Ch. 18 Hyperbola - Chapter Revision Points

Hyperbola - Sections in the R D Sharma Chapter

1. Hyperbola - Definitions

2. Equation of hyperbola in its standard form

3. Second focus and second directrix of the hyperbola

4. Vertices, major and minor axes, foci, directrices and centre of the hyperbola

5. Eccentricity

6. Length of latus rectum

7. Focal distances of a point

8. Conjugate hyperbola

9 .Parametric equations and parametric coordinates

10. Equation of the chord joining any two points on a hyperbola

11. Intersection of a line and a hyperbola

12. Condition of a line to be a tangent to a hyperbola

13. Equation of tangent in different forms

14. Number of tangents drawn from a point to a hyperbola

15. Equation of the pair of tangents from a point to a hyperbola

16. Equation of the chord of contacts of tangents


17. Equation of normal in different forms

18 Number of normals

19. Equation of the chord of a hyperbola bisected at a given point

20 Asymptotes of a hyperbola

21. Rectangular hyperbola

Revision Points

1. Introduction




2. Equation of hyperbola in its standard form

x² /a² - y² /b² = 1

The hyperbola intersects x axis at (a,0) and (-a,0).
The hyperbola does not intersect teh y axis.

x² /a² = 1 +y²/b²
Hence x² /a²≥1
x² ≥a²
x≥a or x≤-a
Hence hyperbola lies on the rightof the line x=a and on the left of the lie x = -a.
the hyperbola consists of two separate branches

Hyperbola is symmetric about both axes.

b² = a²(e²-1)
Where e = eccentricity

Focus is (ae,0)
Directrix is the line x = a/e

Length of latus rectum = 2b²/a







3. Second focus and second directrix of the hyperbola




4. Vertices, major and minor axes, foci, directrices and centre of the hyperbola

At vertices, the curve meets the line joining foci. Vertices for the hyperbola
x² /a² - y² /b² = 1 are (a,0) and (-a,0).

the straight line joining the vertices is called the transverse axis of the hyperbola. Its length is 2a.




5. Eccentricity




6. Length of latus rectum




7. Focal distances of a point




8. Conjugate hyperbola

For the hyperbola x²/a² - y²/b² = 1, the congugate hyperbola is
-x²/a² + y²/b² = 1




9 .Parametric equations and parametric coordinates

x - a sec θ and y = b tan θ are the parametric coordinates

x = a cosh θ and y = b sinh θ are also parametric coordinates.

cosh θ = [e^θ +e^θ]/2
sinh θ = [e^θ - ^θ]/2




10. Equation of the chord joining any two points on a hyperbola

The points are taken as P(a sec θ1,b tan θ1), Q(a sec θ2,b tan θ2)

The equation of the chord joining P and Q is

y -b tan θ1 = [(b tan θ2 - b tan θ1)/(a sec θ2 - a sec θ1)]*(x-a sec θ1)






11. Intersection of a line and a hyperbola



12. Condition of a line to be a tangent to a hyperbola




13. Equation of tangent in different forms





14. Number of tangents drawn from a point to a hyperbola




15. Equation of the pair of tangents from a point to a hyperbola





16. Equation of the chord of contacts of tangents





17. Equation of normal in different forms





18 Number of normals





19. Equation of the chord of a hyperbola bisected at a given point





20 Asymptotes of a hyperbola






21. Rectangular hyperbola


Updated 17 Jan 2016, 5 June 2008