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 e₂ are 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 equation of the auxiliary circle of the hyperbola is given by

x²+ y² = a²

• In parametric form, the equations x = a sec θ and 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

• ?Equation of tangent:

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 = 1 at 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

• Equation 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 combined equation of pair of tangents drawn from a point P(x₁, y₁) lying outside the hyperbola x²/a² – y²/b² = 1 is SS₁ = T², where S = x²/a² – y²/b² - 1, x₁²/a² – y₁²/b² - 1 and T = xx₁/a² – yy₁/b² - 1

• The tangent and normal at any point of a hyperbola bisect the angle between the focal radii.

• The portion of the tangent between the point of contact and the directrix subtends a right angle at the corresponding focus.

• 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.

• Comparison between hyperbola and its conjugate hyperbola:?

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

• If the lengths of transverse and conjugate axis are equal, then such a hyperbola is termed to be a rectangular hyperbola.

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

• The equation xy = c² represents a rectangular hyperbola

• 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

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

• Equation of a chord joining the points P(t₁) and Q(t₂) is x + t₁t₂y = c(t₁ + t₂) with slope m = -1/ t₁t₂

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

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

• Equation of normal is y - c/t = t²(x – ct)

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

• The equation of the director circle of the hyperbola is given by x² + y² = a² - b².

• If b² < a², this circle is real.

• If b² = a² the radius of the circle is zero and it reduces to a point circle at the origin. In this case, centre is the only point from which the tangents at right angles can be drawn to the curve.

• If b² > a² the radius of the circle is imaginary, so that there is no such circle and so no tangents at right angle can be drawn to the curve.