Revision Notes on Hyperbola

• The standard equation of hyperbola with reference to its principal axis along the coordinate axis is given by x²/a² - y²/b² = 1, where b² = a² (e² -1)

• The foci of the hyperbola are S(ae, 0) and S’ = (-ae, 0)

• Equations of the directrices are given by x = a/e and x = -a/e

• The coordinates of vertices are A’ = (-a, 0) and A = (a,0)

• The length of latus rectum is 2b²/a = 2a(e² - 1)

• The length of the transverse axis of the hyperbola is 2a.

• The difference of the focal distances of any point on the hyperbola is constant and is equal to transverse axis.

• x²/a² - y²/b² = 1 and -x²/a² + y²/b² = 1 are conjugate hyperbola of each other.

• If e₁ and e₂are the eccentricities of the hyperbola and its conjugate then e₁^-2+ e₂^-2 = 1

• The foci of a hyperbola and its conjugate are concyclic and form the vertices of a square.

• The length of the transverse axis of a hyperbola is 2a and the transverse axis and conjugate axis together constitute the principal axis of the hyperbola.

• Whether two hyperbolas are similar or not is decided on the basis of their eccentricity. The hyperbolas with same eccentricity are same.

• The eccentricity of rectangular hyperbola is √2 and the length of its latus rectum is equal to its transverse or conjugate axis. 

• The equation of the auxiliary circle of the hyperbola is given by x²+ y² = a²

• In parametric form, the equations x = a sec θand  y = b tan θ together represent the hyperbola x²/a² - y²/b² = 1. Here θ is a parameter.

• The point P(x₁, y₁) lies within, on or outside the ellipse according as x₁²/a² - y₁²/b² = 1 is positive, zero or negative.

• The line y = mx + c is a secant, a tangent or passes outside the hyperbola  x²/a² - y²/b² = 1 according as whether c² is > = or < a²m² - b²

• Equations of tangents

1. Equation of tangent to hyperbola x²/a² - y²/b² = 1at the point (x₁, y₁) is xx₁/a² - yy₁/b² = 1

2. Equation of tangent to hyperbola x²/a² - y²/b² = 1at the point (a sec θ, b tan θ) is (x sec θ)/a - (y tan θ)/b= 1

3. y = mx ±[(a²m²- b²] can also be taken as the tangent to the hyperbola x²/a² - y²/b² = 1

• Equations of normal

1. Equation of normal to the hyperbola x²/a² - y²/b² = 1 at the point (x₁, y₁) is a² x/x₁+ b² y/y₁ = a² - b² = a²e²

2. Equation of normal at the point P(a sec θ, b tan θ) on the hyperbola x²/a² - y²/b² = 1 is ax/sec θ + by/ tan θ = a²+b² = a²e²

• The equation of director circle is x² + y² = a² - b²  

• The equations of the asymptotes of the hyperbola are x/a + y/b = 0 and x/a - y/b = 0. This can be combined as x²/a² - y²/b² = 0

• The asymptotes of the hyperbola and its conjugate are same

• The asymptotes pass through the center of the hyperbola and the bisectors of the angles between the asymptotes are the axis of a hyperbola

• The equation xy = c² represents a rectangular hyperbola

• ü In a hyperbola b² = a² (e² – 1). In the case of rectangular hyperbola (i.e., when b = a) result become a² = a²(e² – 1) or e² = 2 or e =√2

• ü In parametric form, its coordinates are x= ct, y = c/t, t ∈ R ~ {0}

• ü Equation of a chord joining the points P(t₁) and Q(t₂) is

• x + t₁t₂y = c(t₁ + t₂) with slope m = -1/ t₁t₂

1. Equation of tangent at P (x₁, y₁) is x/x₁ + y/y₁ = 2

2. Equation of tangent at P(t) is x/t + ty = 2c

3. Equation of normal is y-c/t = t²(x-ct)

Chord whose middle point is given to be (h, k) is kx + hy = 2hk