Mathematics

Mathematics

Problem

The slope of the tangent to a curve

C: y = y(x)

at any point

(x, y)

on it is

\frac{2e^{2x} - 6e^{-x} + 9}{2 + 9e^{-2x}}

. If

C

passes through the points

\left(0, \frac{1}{2} + \frac{\pi}{2\sqrt{2}}\right)

and

\left(\alpha, \frac{1}{2}e^{2\alpha}\right)

, then

e^\alpha

is equal to [Main July 25, 2022 (I)] (a)

\frac{3+\sqrt{2}}{3-\sqrt{2}}

(b)

\frac{3}{\sqrt{2}} \left( \frac{3+\sqrt{2}}{3-\sqrt{2}} \right)

(c)

\frac{1}{\sqrt{2}} \left( \frac{\sqrt{2}+1}{\sqrt{2}-1} \right)

(d)

\frac{\sqrt{2}+1}{\sqrt{2}-1}

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