Straight Lines
Mirror Line
Grade 11
Question:
<p>Let <span class="math">L_1: 3x + 4y = 1</span> and <span class="math">L_2: 5x - 12y + 2 = 0</span> be two given lines. Let the image of every point on <span class="math">L_1</span> with respect to a line L lies on <span class="math">L_2</span>. Then a possible equation of L can be:</p>
<p>(a) 14x + 112y - 23 = 0</p>
<p>(b) 64x - 8y - 3 = 0</p>
<p>(c) 11x - 4y = 0</p>
<p>(d) 52y - 45x = 7</p>
Step-by-Step Solution
Key Concept: The line L that maps L₁ to L₂ is the angle bisector of L₁ and L₂, passing through their intersection point.
<p><strong>Step 1:</strong> Line L is the mirror line such that <span class="math">L_1</span> and <span class="math">L_2</span> are mirror images across L.</p><p><strong>Step 2:</strong> The mirror line L passes through the intersection of <span class="math">L_1</span> and <span class="math">L_2</span>.</p><p><strong>Step 3:</strong> Find intersection: Solve 3x + 4y = 1 and 5x - 12y + 2 = 0.</p><p><strong>Step 4:</strong> The line L makes equal angles with <span class="math">L_1</span> and <span class="math">L_2</span>.</p><p><strong>Step 5:</strong> Use the angle bisector formula to find L.</p><p>∴ Answer is (b).</p>
Correct Answer: B