Question

From any point P of a rectangular hyperbola x² − y² = a with centre O, the perpendiculars PM and PN are drawn to the principal axes (i.e., usual axes). Show that the tangent at P is perpendicular to MN and that the distance of O from the tangent varies inversely as OP.

Answer 1

Let P =(a secθ, a tanθ) so that M = (a sec θ, 0) and N = (0, a tanθ ). The tangent at P is

x secθ - y tan θ = a

so that the slope of the tangent at P is

sec θ/tan θ = cosec θ

Slope of MN is

-a tanθ/a sec θ =  - sin θ

Now,

Slope of the tangent at P X Slope of MN = cosec θ x (- sin θ) = -1

Hence, the tangent at P is perpendicular to MN. Also d is the distance of O from the tangent at P which is given by