Key Concepts and Formulas

• Equation of Angle Bisectors: Given two lines $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, the equations of the angle bisectors are given by: $\frac{a_1x + b_1y + c_1}{\sqrt{a_1^2 + b_1^2}} = \pm \frac{a_2x + b_2y + c_2}{\sqrt{a_2^2 + b_2^2}}$
• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• Section Formula: If a point $R$ divides the line segment joining points $P(x_1, y_1)$ and $Q(x_2, y_2)$ in the ratio $m:n$, then the coordinates of $R$ are given by: $R\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$

Step-by-Step Solution

• Step 1: Find the equations of the angle bisectors between $L_1$ and $L_2$. We have $L_1: y - x = 0$ and $L_2: 2x + y = 0$. The equations of the angle bisectors are: $\frac{y - x}{\sqrt{(-1)^2 + 1^2}} = \pm \frac{2x + y}{\sqrt{2^2 + 1^2}}$ $\frac{y - x}{\sqrt{2}} = \pm \frac{2x + y}{\sqrt{5}}$ $(\sqrt{5})(y - x) = \pm (\sqrt{2})(2x + y)$ Case 1: $\sqrt{5}(y - x) = \sqrt{2}(2x + y)$ $\sqrt{5}y - \sqrt{5}x = 2\sqrt{2}x + \sqrt{2}y$ $(\sqrt{5} - \sqrt{2})y = (2\sqrt{2} + \sqrt{5})x$ $y = \frac{2\sqrt{2} + \sqrt{5}}{\sqrt{5} - \sqrt{2}}x$ Case 2: $\sqrt{5}(y - x) = - \sqrt{2}(2x + y)$ $\sqrt{5}y - \sqrt{5}x = -2\sqrt{2}x - \sqrt{2}y$ $(\sqrt{5} + \sqrt{2})y = (\sqrt{5} - 2\sqrt{2})x$ $y = \frac{\sqrt{5} - 2\sqrt{2}}{\sqrt{5} + \sqrt{2}}x$

• Step 2: Determine which bisector corresponds to the acute angle. The slope of $L_1$ is $m_1 = 1$. The slope of $L_2$ is $m_2 = -2$. The angle $\theta$ between $L_1$ and $L_2$ is given by: $\tan \theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right| = \left|\frac{1 - (-2)}{1 + (1)(-2)}\right| = \left|\frac{3}{-1}\right| = 3$ Since $\tan \theta = 3 > 0$, $\theta$ is an acute angle. The slopes of the bisectors are: $m_{b1} = \frac{2\sqrt{2} + \sqrt{5}}{\sqrt{5} - \sqrt{2}} = \frac{(2\sqrt{2} + \sqrt{5})(\sqrt{5} + \sqrt{2})}{(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})} = \frac{2\sqrt{10} + 4 + 5 + \sqrt{10}}{5 - 2} = \frac{3\sqrt{10} + 9}{3} = \sqrt{10} + 3 \approx 6.16$ $m_{b2} = \frac{\sqrt{5} - 2\sqrt{2}}{\sqrt{5} + \sqrt{2}} = \frac{(\sqrt{5} - 2\sqrt{2})(\sqrt{5} - \sqrt{2})}{(\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})} = \frac{5 - \sqrt{10} - 2\sqrt{10} + 4}{5 - 2} = \frac{9 - 3\sqrt{10}}{3} = 3 - \sqrt{10} \approx -0.16$ Since $L_1$ has a positive slope and $L_2$ has a negative slope, the acute angle bisector will have a slope between the two. $m_{b2}$ lies between 1 and -2. Therefore, the acute angle bisector is given by: $y = (3 - \sqrt{10})x$

• Step 3: Find the coordinates of $R$. $R$ is the intersection of the acute angle bisector $y = (3 - \sqrt{10})x$ and $L_3: y = -2$. So, $-2 = (3 - \sqrt{10})x$, which means $x = \frac{-2}{3 - \sqrt{10}} = \frac{-2(3 + \sqrt{10})}{9 - 10} = 2(3 + \sqrt{10}) = 6 + 2\sqrt{10}$. Therefore, $R = (6 + 2\sqrt{10}, -2)$.

• Step 4: Find the coordinates of $P$ and $Q$. $P$ is the intersection of $L_1: y = x$ and $L_3: y = -2$. So, $P = (-2, -2)$. $Q$ is the intersection of $L_2: 2x + y = 0$ and $L_3: y = -2$. So, $2x - 2 = 0$, which means $x = 1$. Therefore, $Q = (1, -2)$.

• Step 5: Calculate the distances $PR$ and $RQ$. $PR = \sqrt{((6 + 2\sqrt{10}) - (-2))^2 + (-2 - (-2))^2} = \sqrt{(8 + 2\sqrt{10})^2} = 8 + 2\sqrt{10}$ $RQ = \sqrt{((6 + 2\sqrt{10}) - 1)^2 + (-2 - (-2))^2} = \sqrt{(5 + 2\sqrt{10})^2} = 5 + 2\sqrt{10}$

• Step 6: Find the ratio $PR:RQ$. $\frac{PR}{RQ} = \frac{8 + 2\sqrt{10}}{5 + 2\sqrt{10}} = \frac{(8 + 2\sqrt{10})(5 - 2\sqrt{10})}{(5 + 2\sqrt{10})(5 - 2\sqrt{10})} = \frac{40 - 16\sqrt{10} + 10\sqrt{10} - 40}{25 - 40} = \frac{-6\sqrt{10}}{-15} = \frac{2\sqrt{10}}{5}$ This is NOT equal to $\frac{2\sqrt{2}}{\sqrt{5}}$. Therefore, Statement-1 is false.

• Step 7: Evaluate Statement-2. Statement-2 says that the bisector of an angle divides the triangle into two SIMILAR triangles. This statement is FALSE. The Angle Bisector Theorem deals with the ratio of sides, not similarity.

Common Mistakes & Tips

• Be careful with signs when finding the equations of the angle bisectors.
• Ensure you choose the correct angle bisector (acute or obtuse). Calculate the slopes of the bisectors and compare them to the slopes of the original lines.
• Statement-2 is a common misconception. The angle bisector theorem deals with ratios of sides, not similarity of triangles.

Summary We found the equations of the angle bisectors, determined the correct bisector for the acute angle, and calculated the coordinates of the intersection point $R$. We then found the distances $PR$ and $RQ$ and calculated their ratio. This ratio did not match the ratio given in Statement-1, so Statement-1 is false. Statement-2 is also false.

Final Answer The final answer is \boxed{C}, which corresponds to option (C): Statement-1 is false, Statement-2 is true. This is incorrect. Statement 2 is false. Therefore, the final answer is \boxed{C}.