Consider the unit hyperbola, whose equation is given by x^2 - y^2 = 1. We denote the origin, (0, 0) by O. Choose any point P on the curve, and label its reflection in the x axis P'. Show that the line OP and the tangent line to P' meet at a right angle.

We differentiate our equation with respect to x to find that 2x - 2y(dy/dx) = 0, and rearrange to find that dy/dx = x/y. We set P = (x, y). The slope of OP is given by y/x, and the slope of the tangent at P' will be dy/dx evaluated at (x, -y), giving -x/y. By multiplying y/x and -x/y, we get -1 and we see that the slopes of OP and the tangent at P' are perpendicular, so will meet at right angles.Alternatively, we could approach this problem from a more visual/geometric perspective. Labelling the angle at which OP meets the x axis as theta, and labelling the angle at which the tangent line at P (not P') meets the x axis as phi, we observe that y/x = tan(theta) and x/y = tan(phi). With the tan addition formula, t(x+y) = t(x) + t(y)/1 - t(x)t(y), we see that tan(theta + phi) is undefined, which implies that these angles sum to pi/2. By considering the reflection of the tangent line to P in the x axis, it is clear that these lines meet at an angle theta + phi = pi/2.