xy = 1
⇒ y = 1/x
⇒ dy/dx = -1/x2
⇒ (dy/dx)(a + x = 1, y = 1) = -1/1 = -1
\(\therefore\) slope of tangent of hyperbola xy = 1 at point (1, 1) = m = -1
\(\therefore\) Equation of tangent is
y - y1 = m(x - x1)
⇒ y - 1 = -1(x - 1)
(\(\because\) Tangent is find at point (1, 1))
⇒ y - 1 = 1 - x
⇒ x + y = 2
⇒ x/2 + y/2 = 1
\(\therefore\) a = x - intercept = 2
b = y-intercept = 2
\(\because\) a = b
Hence, the tangent at the point (1, 1) on the rectangular hyperbola xy = 1 cuts equal length of the axes.
Hence Proved.