Straight Lines
Pair of straight lines
Grade 11
Question:
<p>Let \(\frac{2}{3}x^2 + \frac{p}{3}xy + y^2 = (y - mx)(y - m'x)\) and \(-\frac{2}{3}x^2 + \frac{q}{-3}xy + y^2 = \left(y + \frac{1}{m}x\right)(y - m'x)\). Then \(m + m' = -\frac{p}{3}\), \(mm' = \frac{2}{3}\), \(\frac{1}{m} - m' = \frac{-q}{3}\), \(\frac{m'}{m} = -\frac{2}{3}\). Which of the following are correct?</p>
<p>(a) If \(m = 1\), \(m' = 2/3\) and \(p = -5\), \(q = -1\)</p>
<p>(b) If \(m = -1\), \(m' = -2/3\) and \(p = 5\), \(q = 1\)</p>
<p>(c) \(m^2 = 1 \Rightarrow m = \pm 1\)</p>
<p>(d) \(p = 5, q = 1\) when \(m = -1\)</p>
Step-by-Step Solution
Key Concept: Expand both quadratic expressions and compare coefficients of x², xy, and constant terms to establish relationships between slopes m, m', and parameters p, q. The pair of lines representation allows direct coefficient matching.
<p><strong>Step 1: Expand the first equation</strong></p><p>Expand (y - mx)(y - m'x) = y² - m'xy - mxy + mm'x² = y² - (m + m')xy + mm'x²</p><p>Comparing with (2/3)x² + (p/3)xy + y²:</p><p>• Coefficient of x²: mm' = 2/3 ✓ (Statement 2 is CORRECT)</p><p>• Coefficient of xy: -(m + m') = p/3, so m + m' = -p/3 ✓ (Statement 1 is CORRECT)</p><p><strong>Step 2: Expand the second equation</strong></p><p>Expand (y + (1/m)x)(y - m'x) = y² - m'xy + (1/m)xy - (m'/m)x² = y² + (1/m - m')xy - (m'/m)x²</p><p>Comparing with (-2/3)x² + (q/-3)xy + y²:</p><p>• Coefficient of x²: -(m'/m) = -2/3, so m'/m = 2/3 ✗ (Statement 4 claims m'/m = -2/3, which is INCORRECT)</p><p>• Coefficient of xy: (1/m - m') = -q/3 ✓ (Statement 3 is CORRECT)</p><p><strong>Step 3: Verification</strong></p><p>From statements 1 and 2: m + m' = -p/3 and mm' = 2/3</p><p>From statement 4 check: m'/m = 2/3 (NOT -2/3)</p><p>∴ Statements (a), (b), and (c) are correct. Statement (d) is incorrect.</p><p><strong>Note:</strong> If the question asks which statements are correct, the answer is (a), (b), (c). If all four options must be evaluated as correct/incorrect, then (a), (b), (c) are correct and (d) is incorrect.</p>
Correct Answer: (a), (b), (c), and (d)