Area Bounded by Tangent to $xy=c^2$ & x-y Axis

The area of triangle OAB bounded by the tangent to $xy=c^2$ in the 1st quadrant, x-axis & y-axis is (refer figure):

(1) Maximum if P is the midpoint of AB

(2) Increases as P moves downwards or upwards
(3) Constant
(4) Independent of c

Solution

$xy=c^2$ is rectangular hyperbola. The equation of tangent in parametric form at some point P $(ct, \frac {c}{t})$ is given by,

$\frac {x}{t}+yt=2c$

At point A, $x=2ct=OA$

At point B, $y=\frac {2c}{t}=OB$

Area of $\Delta OAB$ = $\frac {1}{2}.OA.OB$ = $\frac {1}{2}.2ct.\frac {2c}{t}$ = $2c^2$

Hence, Option (3).