AP EAPCET 18 May 2024 Shift 1 Solved Paper

Given the problem involves finding the value of $(\alpha + \beta + \gamma)^2$ when $(\alpha, \beta, \gamma)$ are the direction cosines of the angular bisector of two lines with direction ratios $(2,2,1)$ and $(2,-1,-2)$.Finding Direction Cosines:The direction ratios for line $L_1$ are $(2,2,1)$, and for line $L_2$ are $(2,-1,-2)$.The direction cosines for $L_1$ are found as:$\left( \frac{2}{3}, \frac{2}{3}, \frac{1}{3} \right)$Similarly, for $L_2$ the direction cosines are:$\left( \frac{2}{3}, \frac{-1}{3}, \frac{-2}{3} \right)$Perpendicular Check:The dot product of the two direction ratios confirms that the lines are perpendicular:$2 \times 2 + 2 \times (-1) + 1 \times (-2) = 0$Angle Bisector Direction Cosines:The formulas for the direction cosines of the angle bisectors are:$\left( \frac{l_1 + l_2}{2 \cos \frac{\theta}{2}}, \frac{m_1 + m_2}{2 \cos \frac{\theta}{2}}, \frac{n_1 + n_2}{2 \cos \frac{\theta}{2}} \right)$and$\left( \frac{l_1 - l_2}{2 \sin \frac{\theta}{2}}, \frac{m_1 - m_2}{2 \sin \frac{\theta}{2}}, \frac{n_1 - n_2}{2 \sin \frac{\theta}{2}} \right)$Calculate the Direction Cosines:For an angle bisector direction, calculate:$\left( \frac{4}{3 \sqrt{2}}, \frac{1}{3 \sqrt{2}}, \frac{-1}{3 \sqrt{2}} \right)$Another possible set is:$\left( 0, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right)$Calculation of $(\alpha + \beta + \gamma)^2$ :Using $\alpha = \frac{4}{3 \sqrt{2}}, \beta = \frac{1}{3 \sqrt{2}}, \gamma = \frac{-1}{3 \sqrt{2}}$ :Sum:$\alpha + \beta + \gamma = \frac{4}{3 \sqrt{2}} + \frac{1}{3 \sqrt{2}} - \frac{1}{3 \sqrt{2}} = \frac{4}{3 \sqrt{2}}$ $(\alpha + \beta + \gamma)^2 = \left( \frac{4}{3 \sqrt{2}} \right)^2 = \frac{16}{18} = \frac{8}{9}$Alternatively, use $\alpha = 0, \beta = \frac{1}{\sqrt{2}}, \gamma = \frac{1}{\sqrt{2}}$ :Sum:$\alpha + \beta + \gamma = 0 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \sqrt{2}$Squaring:$(\alpha + \beta + \gamma)^2 = (\sqrt{2})^2 = 2$Based on calculations with valid assumptions, the value of $(\alpha + \beta + \gamma)^2$ considering a correct scenario for calculation comes out to be 2 .