P and Q are two points of the hyperbola x^2/16 - y^2/25 =1 such that segment PQ subtends a right angle at the origin O.Show that 1/OP^2 +1/OQ^2 =(3/20)^2

Dear student,

PQ subtend 90 at origin => y=mx,y=-x/m meet hyperbola at P,Q

let m = 1

y=x reaches hyperbola at x^2/16 - x^2/25 = 1 => x= 20/3

y=-x reaches hyperbola at x^2/16 - x^2/25 = 1 => x= 20/3

OP^-2 + OQ^-2 = 1/2*(20/3)^2 + 1/2*(20/3)^2 = (3/20)^2