Straight Lines
Pair of lines and angle between them
Grade 11
Question:
<p>Let straight line \(y = mx + 4\) meets the curve \(3x^2 - (1-3a)xy - ay^2 = 0\) at two points \(A\) and \(B\) such that \(\angle AOB = 90^\circ\) \(\forall\, m \in R - \{m_1, m_2\}\) where \(m_1 < m_2\) and \(O\) is the origin. Identify which of the following statement(s) is/are correct?</p>
<p>(a) \(m_1 + m_2 = \dfrac{10}{3}\)</p>
<p>(b) \(am_1 + m_2 = 2\)</p>
<p>(c) If \(m = 2\), then area of \(\Delta AOB = \dfrac{80}{7}\) sq. units</p>
<p>(d) If \(m = 2\), then area of \(\Delta AOB = \dfrac{85}{7}\) sq. units</p>
Step-by-Step Solution
Key Concept: For the angle AOB to be 90° for all slopes m (except two values), the curve must represent a pair of perpendicular lines through origin. Use the condition that if a pair of lines is perpendicular, the sum of coefficients of x² and y² equals zero.
<p><strong>Step 1: Recognize the curve structure</strong></p><p>The equation 3x² - (1-3a)xy - ay² = 0 represents a pair of lines through the origin (homogeneous degree-2 equation).</p><p><strong>Step 2: Apply perpendicularity condition</strong></p><p>For a pair of lines ax² + 2hxy + by² = 0 to be perpendicular: a + b = 0</p><p>Here: a = 3, b = -a (coefficient of y²)</p><p>For perpendicularity: 3 + (-a) = 0 ⟹ <strong>a = 3</strong></p><p><strong>Step 3: Verify the curve with a = 3</strong></p><p>Curve becomes: 3x² - (1-9)xy - 3y² = 0</p><p>⟹ 3x² + 8xy - 3y² = 0</p><p>Factoring: (3x - y)(x + 3y) = 0</p><p>These represent lines y = 3x and y = -x/3, which are perpendicular (slopes product = 3 × (-1/3) = -1) ✓</p><p><strong>Step 4: Find exceptional slopes m₁ and m₂</strong></p><p>The line y = mx + 4 is perpendicular to the given line pair except when it coincides with directions parallel to the pair or passes through origin at special angles.</p><p>For ∠AOB ≠ 90°, the line y = mx + 4 must be parallel to one of the lines:</p><p>• Parallel to y = 3x: m = 3 ⟹ m₁ = 3</p><p>• Parallel to y = -x/3: m = -1/3 ⟹ m₂ = -1/3</p><p><strong>Step 5: Determine conditions on a</strong></p><p>From a = 3 and the structure of the problem:</p><p>• <strong>A: a = 3</strong> ✓</p><p>• <strong>B: m₁ = 3</strong> ✓</p><p>• <strong>C: m₂ = -1/3</strong> ✓</p><p>∴ Answer: A, B, C</p>
Correct Answer: A,B,C