<p>Let <em>y = m<sub>1</sub>x</em> and <em>y = m<sub>2</sub>x</em> be two straight lines represented by <em>x</em><sup>2</sup> − 2<em>cxy</em> − 7<em>y</em><sup>2</sup> = 0. If <em>(m<sub>1</sub> + m<sub>2</sub>) = 4m<sub>1</sub>m<sub>2</sub></em>, then the value of <em>c</em> is:</p>
Step-by-Step Solution
Key Concept: For a pair of lines ax² + 2hxy + by² = 0 passing through origin, use Vieta's formulas: sum of slopes = -2h/b and product of slopes = a/b. Apply the given constraint to find the unknown parameter.
<p><strong>Step 1:</strong> Compare x² − 2cxy − 7y² = 0 with standard form ax² + 2hxy + by² = 0</p><p>Here: a = 1, 2h = −2c (so h = −c), b = −7</p><p><strong>Step 2:</strong> For pair of lines y = m₁x and y = m₂x, apply Vieta's formulas:</p><p>• Sum of slopes: m₁ + m₂ = −2h/b = −(−2c)/(−7) = −2c/7</p><p>• Product of slopes: m₁m₂ = a/b = 1/(−7) = −1/7</p><p><strong>Step 3:</strong> Use given condition m₁ + m₂ = 4m₁m₂</p><p>−2c/7 = 4(−1/7)</p><p>−2c/7 = −4/7</p><p>−2c = −4</p><p>c = 2</p><p><strong>∴ Answer: C (c = 2)</strong></p>
Correct Answer: C