70) A tangent line is drawn to the hyperbola xy = c at a point P.
a) Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is P.
b) 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.
For a) -
y = c/x, so y' = -c/x^2
Let the coordinates at point P be (a,b)
y - b = (-c/a^2)(x - a) then is the equation of the tangent line at point P.
Then, using distributive property and b = c/a, it simplifies to:
y = c(2a - x)/a^2
Next, find the x and y intercepts.
The y intercept is (0, 2c/a). The x intercept is (c(2a - x)/a^2,0).
Using the midpoint formula, the midpoint between the x intercept and y intercept is [c(2a-x)/2a^2,c/a]. The y value looks fine (b = c/a), but shouldn’t the x value be a? Where did I mess up?
For b) do I just calculate the length between the x and y intercepts, and then use the Pythagorean theorem to calculate the area? How do I prove that the area is the same for every point P? Do I need to eliminate a and b from the area somehow?