Question

If \( L_1 : x - y = 0 \), \( L_2 : y = -3x \), \( L_3 \) is the obtuse angle bisector of \( L_1 \) and \( L_2 \), and \( L_4 : x + 3 = 0 \). Let \( A \) be the point of intersection of \( L_4 \) and \( L_1 \), \( B \) be the point of intersection of \( L_4 \) and \( L_2 \), and \( C \) be the point of intersection of \( L_4 \) and \( L_3 \). Then the value of \( \dfrac{(BC)^2}{(AC)^2} \) is:

• A. 4
• B. 5
• C. 6
• D. 7

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The bisector of an angle between two lines $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ is $\frac{a_1x + b_1y + c_1}{\sqrt{a_1^2 + b_1^2}} = \pm \frac{a_2x + b_2y + c_2}{\sqrt{a_2^2 + b_2^2}}$.
If $a_1a_2 + b_1b_2>0$, the '+' sign gives the obtuse bisector (provided constants are made positive).
Step 2: Key Formula or Approach:
1. Find the equation of the obtuse bisector $L_3$.
2. Find coordinates of $A, B, C$ by intersecting with $x = -3$.
3. Calculate distance ratios.
Step 3: Detailed Explanation:
1. Line equations: $L_1: x - y = 0$, $L_2: 3x + y = 0$.
$a_1a_2 + b_1b_2 = (1)(3) + (-1)(1) = 2$.
Since it's positive, the '+' sign in $\frac{x - y}{\sqrt{1^2 + (-1)^2}} = \pm \frac{3x + y}{\sqrt{3^2 + 1^2}}$ gives the obtuse bisector.
\[ \frac{x - y}{\sqrt{2}} = \frac{3x + y}{\sqrt{10}} \implies \sqrt{5}(x - y) = 3x + y \implies (\sqrt{5} - 3)x - (\sqrt{5} + 1)y = 0 \]
2. Finding coordinates on $x = -3$:
$A (L_1 \cap L_4): y = x = -3 \implies A(-3, -3)$.
$B (L_2 \cap L_4): y = -3(-3) = 9 \implies B(-3, 9)$.
$C (L_3 \cap L_4): (\sqrt{5} - 3)(-3) = (\sqrt{5} + 1)y \implies y = \frac{-3(\sqrt{5} - 3)}{\sqrt{5} + 1}$.
Rationalize $y_C$: $\frac{-3(\sqrt{5} - 3)(\sqrt{5} - 1)}{5 - 1} = \frac{-3(5 - \sqrt{5} - 3\sqrt{5} + 3)}{4} = \frac{-3(8 - 4\sqrt{5})}{4} = -6 + 3\sqrt{5}$.
3. Distance ratios: Since all points lie on $x=-3$, distance is just diff of $y$.
$y_A = -3, y_B = 9, y_C = 3\sqrt{5} - 6$.
$BC = |9 - (3\sqrt{5} - 6)| = 15 - 3\sqrt{5}$.
$AC = |(3\sqrt{5} - 6) - (-3)| = 3\sqrt{5} - 3$.
Ratio $BC/AC = \frac{3(5 - \sqrt{5})}{3(\sqrt{5} - 1)} = \frac{5 - \sqrt{5}}{\sqrt{5} - 1} = \frac{\sqrt{5}(\sqrt{5} - 1)}{\sqrt{5} - 1} = \sqrt{5}$.
Squared ratio: $(\sqrt{5})^2 = 5$.
Step 4: Final Answer:
The value is 5.