Equation of a common tangent to the curves y2 = 8x and xy = – 1 is

Equation of a tangent at at2,2at  to y2=8xty=x+at2  where 4a=8 i.e. a=2⇒ ty=x+2t2 which intersects the curve xy=−1 at the points given by xx+2t2t=−1 clearly t≠0 or x2+2t2x+t=0 and will be a tangent to the curve if the roots of this quadratic equation are equal, for which 4t4 – 4t = 0⇒ t = 0 or t = 1 and an equation of a common tangent is y = x + 2.