Straight Lines
Pair of Lines
Grade 11
Question:
<p>The lines <span class="math">L_1</span> and <span class="math">L_2</span> denoted by <span class="math">3x^2 + 10xy + 8y^2 + 14x + 22y + 15 = 0</span> intersect at the point P and have gradients <span class="math">m_1</span> and <span class="math">m_2</span> respectively. The acute angle between them is <span class="math">\theta</span>. Which of the following relations hold good:</p>
<p>(a) <span class="math">m_1 + m_2 = \frac{5}{4}</span></p>
<p>(b) <span class="math">m_1 m_2 = \frac{3}{8}</span></p>
<p>(c) <span class="math">\theta = \sin^{-1}\left(\frac{2}{5\sqrt{5}}\right)</span></p>
<p>(d) sum of the abscissa and ordinate of point P is –1</p>
Step-by-Step Solution
Key Concept: A second-degree equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 represents a pair of lines. Use the condition that Δ = 0 (determinant condition) and decompose the quadratic form to find gradients, point of intersection, and angle between lines.
<p><strong>Step 1: Rewrite in standard form</strong></p><p>Given: 3x² + 10xy + 8y² + 14x + 22y + 15 = 0</p><p>In the form ax² + 2hxy + by² + 2gx + 2fy + c = 0:</p><p>a = 3, 2h = 10 → h = 5, b = 8, 2g = 14 → g = 7, 2f = 22 → f = 11, c = 15</p><p><strong>Step 2: Verify it represents a pair of lines</strong></p><p>Check Δ = abc + 2fgh - af² - bg² - ch²</p><p>Δ = 3(8)(15) + 2(11)(7)(5) - 3(121) - 8(49) - 15(25)</p><p>= 360 + 770 - 363 - 392 - 375 = 0 ✓</p><p><strong>Step 3: Find m₁ + m₂</strong></p><p>For pair of lines: m₁ + m₂ = -2h/b = -10/8 = -5/4</p><p>Option (a) claims m₁ + m₂ = 5/4, which is FALSE.</p><p><strong>Step 4: Find m₁m₂</strong></p><p>m₁m₂ = a/b = 3/8</p><p>Option (b) is TRUE.</p><p><strong>Step 5: Find the angle between lines</strong></p><p>tan θ = |2√(h² - ab)|/(a + b) = |2√(25 - 24)|/(3 + 8) = |2√1|/11 = 2/11</p><p>Also, sin θ = |2√(h² - ab)|/√[(a+b)² + 4(h² - ab)]</p><p>= 2/√[121 + 4] = 2/√125 = 2/(5√5)</p><p>Option (c) is TRUE.</p><p><strong>Step 6: Find point of intersection P</strong></p><p>Point P satisfies: ∂f/∂x = 0 and ∂f/∂y = 0</p><p>6x + 10y + 14 = 0 → 3x + 5y + 7 = 0 ... (i)</p><p>10x + 16y + 22 = 0 → 5x + 8y + 11 = 0 ... (ii)</p><p>From (i): 3x + 5y = -7</p><p>From (ii): 5x + 8y = -11</p><p>Multiply (i) by 5: 15x + 25y = -35</p><p>Multiply (ii) by 3: 15x + 24y = -33</p><p>Subtract: y = -2, then 3x + 5(-2) = -7 → 3x = 3 → x = 1</p><p>Therefore P = (1, -2)</p><p>Sum of abscissa and ordinate = 1 + (-2) = -1</p><p>Option (d) is TRUE.</p><p><strong>∴ Answer:</strong> Multiple</p>
Correct Answer: Multiple