Question

{"given":"The equation $x^2 - 2cxy - 7y^2 = 0$ represents a pair of straight lines passing through the origin. We need to find the value of $c$ when the sum of slopes equals four times their product. ","key_observation":"For a homogeneous equation of second degree $ax^2 + 2hxy + by^2 = 0$, if it represents a pair of lines with slopes $m_1$ and $m_2$, then $m_1 + m_2 = -\\frac{2h}{b}$ and $m_1 \\cdot m_2 = \\frac{a}{b}$. The condition given is $m_1 + m_2 = 4(m_1 \\cdot m_2)$.","option_analysis":[{"label":"(A)","text":"1","verdict":"incorrect","explanation":"If $c = 1$, then sum of slopes = $\\frac{2}{7}$ and product = $-\\frac{1}{7}$. Check: $\\frac{2}{7} = 4 \\times (-\\frac{1}{7}) = -\\frac{4}{7}$, which is false."},{"label":"(B)","text":"-1","verdict":"incorrect","explanation":"If $c = -1$, then sum of slopes = $-\\frac{2}{7}$ and product = $-\\frac{1}{7}$. Check: $-\\frac{2}{7} = 4 \\times (-\\frac{1}{7}) = -\\frac{4}{7}$, which gives $-\\frac{2}{7} = -\\frac{4}{7}$, which is false."},{"label":"(C)","text":"2","verdict":"correct","explanation":"Comparing with $ax^2 + 2hxy + by^2 = 0$: $a = 1$, $2h = -2c$ so $h = -c$, $b = -7$. Sum of slopes = $-\\frac{2h}{b} = -\\frac{2(-c)}{-7} = -\\frac{2c}{7}$. Product = $\\frac{a}{b} = \\frac{1}{-7} = -\\frac{1}{7}$. Given condition: $-\\frac{2c}{7} = 4 \\times (-\\frac{1}{7}) = -\\frac{4}{7}$. Therefore $-2c = -4$, so $c = 2$."},{"label":"(D)","text":"-2","verdict":"incorrect","explanation":"If $c = -2$, then sum of slopes = $\\frac{4}{7}$ and product = $-\\frac{1}{7}$. Check: $\\frac{4}{7} = 4 \\times (-\\frac{1}{7}) = -\\frac{4}{7}$, which is false since $\\frac{4}{7} \\neq -\\frac{4}{7}$."}],"answer":"(C)","formula_steps":[]}