calculus- tangent line of hyperbola?

Question

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

Answer 1

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 coordinate axes is P.

Let P(a, c/a).

xy = c
y = c/x

dy/dx = -c/x²

Now write the equation of the tangent line thru P.

y - c/a = (-c/a²)(x - a) = -cx/a² + c/a
y = -cx/a² + 2c/a

The x and y intercepts are (2a, 0) and (0, 2c/a). Note that the x and y intercepts are twice that of point P(a, c/a). Therefore P is always the midpoint of the tangent line in the first quadrant.
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b) Show that the triangle formed by the tangent line and coordinate axes alywas has the same area, no matter where P is located on the hyperbola.

Area tri = (1/2)(2a)(2c/a) = 2c

for any point P.