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Question

The abscissa if points $P blank$ and $Q$ on the curve $y equals e to the power of x end exponent plus blank e to the power of negative x end exponent$ such that tangents at $P$ and $Q$ make $60$ with the $x$ -axis

In $(\frac{\sqrt{3} + \sqrt{7}}{7})$ and in $(\frac{\sqrt{3} + \sqrt{5}}{2})$
In $(\frac{\sqrt{3} + \sqrt{7}}{2})$
In $(\frac{\sqrt{7} - \sqrt{3}}{2})$
$\pm$ In $(\frac{\sqrt{3} + \sqrt{7}}{2})$

The correct answer is: In $open parentheses fraction numerator square root of 3 plus square root of 7 over denominator 2 end fraction close parentheses$

$y equals e to the power of x end exponent plus blank e to the power of negative x end exponent rightwards double arrow fraction numerator d y over denominator d x end fraction equals e to the power of x end exponent minus blank e to the power of negative x end exponent equals tan invisible function application theta$ , where $theta$ is the angle of the tangent with the $x$ -axis
For $theta equals 60 degree$ , we have $tan invisible function application 60 equals e to the power of x end exponent minus blank e to the power of negative x end exponent$
$rightwards double arrow e to the power of 2 x end exponent minus blank square root of 3 e to the power of x end exponent minus 1 equals 0$
$rightwards double arrow e to the power of x end exponent equals fraction numerator square root of 3 plus-or-minus square root of 7 over denominator 2 end fraction rightwards double arrow x equals log subscript e end subscript invisible function application open parentheses fraction numerator square root of 3 plus square root of 7 over denominator 2 end fraction close parentheses$

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