Question

please help, i procrastinated and now this is due tomorrow!!


A tangent line is drawn to the hyerbola xy=c at a point P.

1) show that the midpoint of the line segment cut from the tangent line by the coordinate axes is P.

2) show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where P is located on the hyperbola.

Answer 1

let p(a,c/a) be the point on the hyperbola

for xy=c
dy/dx = -y/x, so at P the slope = -c/a^2

equation of tangent line:
y - c/a = -c/a^2(x - a) which when simplified is
cx - a^2y=-2ac

for x-intercept, let y=0, then x = 2a
for y-intercept, let x=0, then y = 2c/a

1. take the midpoint of (0,2c/a) and (2a,0) and what do you get????

2. aren't your x and y intercepts the base and height of your triangle??
take area = 1/2 base*height
= .....
= c which is the constant of the original equation!!!