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### 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).

### Sum of the coefficients in the expansion of $(x+y)^n$ ....

If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096, then the greatest coefficient in the expansion is _ _ _ _ . Solution $C_0 + C_1 + C_2 + C_3 + ......................... + C_n =4096$ $\therefore 2^n = 4096 =2^{12}$ $\Rightarrow n = 12$ Greatest coefficient = ${}^{12}{C_6} = 924$