find the eq of the directrix and foci of the rectangular hyperbola xy=c2

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

i.e. the eccentricity of a rectangular hyperbola = √2.

• In case of rectangular hyperbola a = b i.e., the length of transverse axis = length of conjugate axis.

• A rectangular hyperbola is also known as an equilateral hyperbola.

• The asymptotes of rectangular hyperbola are y = ± x.

• If the axes of the hyperbola are rotated by an angle of -π/4 about the same origin, then the equation of the rectangular hyperbola x2 – y2 = a2 is reduced to xy = a2/2 or xy = c2.

• When xy = c2, the asymptotes are the coordinate axis.

• Length of latus rectum of rectangular hyperbola is the same as the transverse or conjugate axis.

• Rectangular Hyperbola with asymptotes as coordinate axis:

• The equation of the hyperbola which has its asymptotes as the coordinate axis is xy = c2 with parametric representation x = ct and y = c/t, t ∈ R-{0}.

• The equations of the directrices of the hyperbola in this case are x + y = ± √2c.

• Since, the transverse and the conjugate axes are the same hence, length of latus rectum = 2√2c = T.A. = C.A.

• Equation of a chord whose middle point is given to be (p, q) is qx + py = 2pq.

• The equation of the tangent at the point P(x1, y1) is x/x1 + y/y1 = 2 and at P(t) is x/t + ty = 2c.

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

• The equation of the chord joining the points P(t1) and Q(t2) is x + t1t2y = c(t1 + t2) and its slope is m = -1/t1t2.  

• The vertices of the hyperbola are (c, c) and (-c, -c) and the focus is (√2c, √2c) and (-√2c, -√2c).