Problem 87 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The lines \(L_{1}\) : \(y-x=0\) and \(L_{2}: 2 x+y=0\) intersect the line \(L_{3}: y+2=0\) at two respective points \(P\) and \(Q\). The bisector of the acute angle between \(L_{1}\) and \(L_{2}\) intersect \(L_{3}\) at \(R\) [2011] Statement - 1 : The ratio \(P R: R Q\) equals \(2 \sqrt{2}: \sqrt{5}\). Statement - 2 : In any triangle, bisector of an angle divides the triangle into two similar triangles. (A) Statement \(-1\) is true, Statement- 2 is true; Statement \(-2\) is not a correct explanation for Statement \(-1\) (B) Statement - 1 is true, Statement- 2 is false. (C) Statement - 1 is false, Statement- 2 is true. (D) Statement - 1 is true, Statement \(-2\) is true; Statement \- 2 is a correct explanation for Statement \(-1\)

Step-by-Step Solution

Answer

Statement 1 is false, Statement 2 is true. (Option C)

Step 1: Find Intersection Points

First, find the points of intersection of lines \(L_1\) and \(L_2\) with line \(L_3\). The line \(L_3\) given by \(y + 2 = 0\) means \(y = -2\). For \(L_1: y - x = 0\), substitute \(y = -2\), giving \(-2 - x = 0 \Rightarrow x = -2\). Thus, \(P(-2, -2)\). For \(L_2: 2x + y = 0\), substitute \(y = -2\), giving \(2x - 2 = 0 \Rightarrow x = 1\). Thus, \(Q(1, -2)\).

Step 2: Determine the Angle Bisector

Find the line representing the angle bisector of the angle formed by lines \(L_1\) and \(L_2\). The angle bisector formula for two lines \(y = m_1x + c_1\) and \(y = m_2x + c_2\) is \[ y - m_1x = \pm \frac{\sqrt{1+m_1^2}}{\sqrt{1+m_2^2}}(y-m_2x).\]Substituting \(m_1 = 1\) and \(m_2 = -2\) gives the bisector equation as \[y = \pm \frac{1}{\sqrt{5}}(x + 2y).\]Choosing the acute angle bisector, we derive \(y = \frac{1}{3}(x + 2y)\). Simplifying, the equation becomes \(x - y = 0\), which is equivalent to \(x = y\).

Step 3: Find Intersection with Line L3

Determine where the angle bisector intersects \(L_3\): \(x = y\) intersects \(y + 2 = 0\) at \(y = -2\). Thus, substituting back we find \(x = -2\) when \(y = -2\), giving point \(R(-2, -2)\).

Step 4: Calculate the Ratio PR:RQ

Calculate the distance from point \(P(-2, -2)\) to \(R(-2, -2)\), and from \(R(-2, -2)\) to \(Q(1, -2)\). For \(P\) and \(R\): the distance \(PR = 0\) (since they coincide). For \(R\) and \(Q\): the distance is \( \sqrt{(1 - (-2))^2 + ((-2) - (-2))^2} = 3\).The ratio \(PR:RQ = 0:3\). The statement (2) doesn't hold as bisectors create a ratio only if there's a triangle. No valid triangle formed due to overlap.

Key Concepts

Lines and Angles

Understanding how lines and angles work is crucial. Lines are straight paths extending in both directions without end. When two lines intersect, they form angles. In the exercise, line \(L_1\) is represented by the equation \(y - x = 0\), and line \(L_2\) by \(2x + y = 0\). These lines intersect another line, \(L_3: y + 2 = 0\).
Lines \(L_1\) and \(L_2\) meet at different points on this line. The intersection of lines can form various angles depending on their slopes. The angle bisector divides the angle into two equal parts. This concept is crucial in calculating relative angles and understanding angles' properties.
The angle bisector of lines \(L_1\) and \(L_2\) helps in identifying critical points needed to solve real-life geometric problems. It is vital in situations where determining equal angles is necessary.

Intersection Points

Finding intersection points is where geometry meets algebra. To find where these lines intersect, set their equations equal to each other. The line equation \(L_3: y + 2 = 0\) means that for any intersection with this line, the \(y\) value is always \(-2\).
For line \(L_1: y - x = 0\), place \(y = -2\) to find \(x\). It results in \(x = -2\), giving intersection point \(P(-2, -2)\). Similarly, for \(L_2: 2x + y = 0\), by substituting \(y = -2\), we derive \(x = 1\), thus point \(Q(1, -2)\).
Intersection points are foundational in analyzing geometrical figures, helping solve tangible problems where paths meet. They are essential in solving many mathematical problems involving distances and angles.

Ratio Calculation

Calculating ratios helps in comparing segments and parts. Ratios express the quantitative relation between two amounts, showing the number of times one value contains or is contained within the other.
In this problem, after determining the intersection points, the goal is to find the ratio \(PR: RQ\) with points \(P(-2, -2)\), \(R(-2, -2)\), and \(Q(1, -2)\).
The distance \(PR\) is \(0\) because \(P\) and \(R\) overlap. While \(RQ = 3\), indicating that the line segment from \(R\) to \(Q\) is 3 units long.
The calculation highlights why Statement 1 can't be correct. It suggests ratios created from bisected angles are not valid if the intersection does not produce distinct segments. Understanding the conditions under which ratios apply is critical for aligning geometric principles with algebraic results.