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For the pair of lines represented by x² - 2c*x*y - 7y² = 0, the sum of the slopes of the two lines equals four times their product. Find c.
- -2
- -1
- 2
- 1
Correct answer: 2
Solution
For the homogeneous equation x² - 2c*x*y - 7y² = 0, dividing by y²: (x/y)² - 2c*(x/y) - 7 = 0. Let m = x/y (but note slopes of the lines are y/x if we write y = mx, so slopes are roots of 7m² + 2cm - 1 = 0 after substituting x = my). Actually: y = mx => x² - 2c*x*(mx) - 7(mx)² = 0 is wrong. Let's substitute y = mx into the equation treating x as parameter: x² - 2c*x*(mx) - 7m²x² = 0 => 1 - 2cm - 7m² = 0 => 7m² + 2cm - 1 = 0. Slopes m1+m2 = -2c/7, m1*m2 = -1/7. Condition m1+m2 = 4*m1*m2: -2c/7 = 4*(-1/7) = -4/7 => -2c = -4 => c = 2.
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