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Let P&Q are two points on the curve $y = \log_{\frac{1}{2}} (x - 0.5) + \log_{2} \sqrt{4 x^{2} - 4 x + 1}$ and P is also on the circle $x^{2} + y^{2} = 10,$ Q lies inside the given circle such that its abscissa is an integer then the smallest possible value of $\vec{O P .} \vec{O Q}$ Where ‘O’ begin origin___

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Detailed Solution

$y = \log_{\frac{1}{2}} (x - 0.5) + \log_{2} \sqrt{4 x^{2} - 4 x + 1}$

$y = - \log_{2} (x - \frac{1}{2}) + \log_{2} 2 \sqrt{x^{2} - x + \frac{1}{4}}$

$\log_{2} [{\frac{2 \sqrt{(x - \frac{1}{2})}}{x - \frac{1}{2}}}^{2}] = 1$

$y = 1$

$p = (x_{1}, 1) Q = (x_{2}, 1)$

$x^{2} + y^{2} = 10, x^{2} + 1 = 10$

$x = \pm 3$

$P (3, 1)$

$\begin{array}{l} \vec{O P} = 3 i + \hat{j} \\ \vec{O Q} = i + \hat{j}, 2 i + \hat{j} \\ \vec{O P} . \vec{O Q} = 4 \\ \vec{O P} . \vec{O Q} = 7 \end{array}$

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